Solutions of Heun’s general equation and elliptic Darboux equation
(An earlier version in: Math. Meth. Appl. Sci., 44, 7165-7206 (2021))

Bartolomeu D. B. Figueiredo<EMAIL_ADDRESS>
Centro Brasileiro de Pesquisas Físicas,

Rua Dr. Xavier Sigaud, 150, CEP 22290-180, Rio de Janeiro, RJ, Brasil

###### Abstract

New solutions for the elliptic Darboux equation are obtained as particular
cases of solutions constructed for Heun’s general equation. We consider two
groups of power series expansions and two new groups of expansions in series
of Gauss hypergeometric functions. The convergence of one group in power
series is determined by means of ratio tests for infinite series, while the
other groups are designed to solve problems which admit finite-series
solutions. Actually, we envisage periodic quasi-exactly solvable potentials
for which the stationary one-dimensional Schrödinger equation is reduced to
the Darboux equation. In general, finite- and infinite-series solutions are
obtained from power series expansions for Heun’s equation. However, we show
that the Schrödinger equation admits additional finite-series expansions in
terms of hypergeometric functions for a family of associated Lamé potentials
used in band theory of solids. For each finite-series solution we find as well
four infinite-series expansions which are bounded and convergent for all
values of the independent variable. Finally we find that it is possible to use
transformations of variables in order to generate new solutions for the
Darboux equation out of other expansions in series of hypergeometric functions
for the Heun equation.

###### Contents

  1. 1 Introduction
  2. 2 Mathematical Preliminaries
    1. 2.1 Elliptic Darboux equation and the Schrödinger equation
    2. 2.2 The Gauss hypergeometric function
    3. 2.3 Some transformations of Heun’s general equation
    4. 2.4 Three-term recurrence relations and convergence
  3. 3 Power-series solutions
    1. 3.1 Series $\mathring{H}^{(i)}(x)$ around $x=0$
    2. 3.2 Series $\mathring{\bm{H}}^{(i)}(x)$ around $x=1$
  4. 4 Solutions in series of hypergeometric functions
    1. 4.1 The initial solution $\bar{H}^{(1)}(x)$
    2. 4.2 The solutions ${H}^{(i)}(x)$ around ${(a-1)x}/{(a-x)}=0$
    3. 4.3 The solutions ${\bm{H}}^{(i)}(x)$ around $x=1$
  5. 5 Schrödinger equation for associated Lamé potentials
    1. 5.1 Finite-series eigenfunctions $\left(\mathring{\psi}^{(i)},\tilde{\psi}^{(i)}\right)$ in terms of $\mathrm{sn}^{2}u$
      1. 5.1.1 First type: Even solutions, period $2K$
      2. 5.1.2 Second type: Odd solutions, period $4K$
      3. 5.1.3 Third type: Even solutions, period $4K$
      4. 5.1.4 Fourth type: Odd solutions, period $2K$
    2. 5.2 Finite-series eigenfunction $\left(\mathring{\bm{\Psi}}^{(i)},\tilde{\bm{\Psi}}^{(i)}\right)$ in terms of $\mathrm{cn}^{2}u$
      1. 5.2.1 First type: even solutions, period $2K$
      2. 5.2.2 Second type: odd solutions, period $4K$
      3. 5.2.3 Third type: even solutions, period $4K$
      4. 5.2.4 Fourth type: odd solutions, period $2K$
    3. 5.3 Finite series of hypergeometric functions ${\psi}^{(i)}(u)$ when $\bm{l}$ is an integer and $\bm{m}\neq-1/2$ is half an odd integer
      1. 5.3.1 Finite series when $\bm{l}$ is an integer and $\bm{m}$ is half an odd positive integer
      2. 5.3.2 Finite series when $\bm{l}$ is an integer and $\bm{m}\neq-1/2$ is half an odd negative integer
    4. 5.4 Finite series $\bm{\Psi}^{(i)}(u)$ of hypergeometric functions when $\bm{m}$ is an integer and $\bm{l}\neq-1/2$ is half an odd integer
      1. 5.4.1 Finite series when $\bm{m}$ is an integer and $\bm{l}$ is half an odd positive integer
      2. 5.4.2 Finite series when $\bm{m}$ is an integer and $\bm{l}\neq-1/2$ is half an odd negative integer
    5. 5.5 Infinite-series solutions
  6. 6 Conclusions
  7. A Homotopic transformations
  8. B Cases solved by hypergeometric functions
  9. C Some degenerate power-series solutions
  10. D Svartholm’s and Erdelyi’s solutions

##  1 Introduction

We find new solutions for the elliptic Darboux equation from solutions of
Heun’s general equation since this includes the former equation as a
particular case. By using known substitutions of variables for Heun’s
equation, we construct two groups of power series expansions and two groups of
expansions in series of Gauss hypergeometric functions. Finally, we find that
these groups are sufficient to solve systematically the Darboux equations
which emerge from the stationary one-dimensional Schrödinger equation for some
periodic quasi-exactly solvable (QES) potentials.

The elliptic form of the Darboux equation can be written as [1, 2]

$\begin{array}[]{l}\frac{d^{2}U(u)}{du^{2}}+\left[h-\mu(\mu+1)k^{2}{\rm
sn}^{2}u-\frac{\nu_{1}\left(\nu_{1}+1\right)}{{\rm
sn^{2}}u}-\frac{\nu_{2}\left(\nu_{2}+1\right){\rm dn^{2}}u}{{\rm
cn^{2}}u}-\frac{\lambda\left(\lambda+1\right)k^{2}{\rm cn}^{2}u}{{\rm
dn^{2}}u}\right]U(u)=0,\end{array}$ (1)

where $h$, $\mu$, $\nu_{1}$, $\nu_{2}$ and $\lambda$ are constants, while
$\operatorname{sn}{u}$, $\operatorname{cn}{u}$ and $\operatorname{dn}{u}$ are
the Jacobian elliptic functions [3] modulus $k$ ($0<k^{2}<1$). Eq. (1) reduces
to the associated Lamé equation when
$\nu_{1}(\nu_{1}+1)=\nu_{2}(\nu_{2}+1)=0$, and to the Lamé equation when
$\nu_{1}(\nu_{1}+1)$$=\nu_{2}(\nu_{2}+1)=$$\lambda(\lambda+1)=0$.

The Darboux equation is a particular case of the Heun equation which, in an
algebraic form [4, 5], reads

$\begin{array}[]{l}\frac{d^{2}H(x)}{dx^{2}}+\left[\frac{\gamma}{x}+\frac{\delta}{x-1}+\frac{\epsilon}{x-a}\right]\frac{dH(x)}{dx}+\frac{\alpha\beta
x-q}{x(x-1)(x-a)}H(x)=0,\quad\epsilon=\alpha+\beta+1-\gamma-\delta,\end{array}$
(2)

where $a\in\mathbb{C}\setminus\\{0,1\\}$, and the points $x=0,1,a,\infty$ are
regular singularities. If $a=0$ or $a=1$ the above equation can be reduced to
the Gauss hypergeometric equation [6]

$\displaystyle\begin{array}[]{l}z(1-z)\frac{d^{2}Z(z)}{dz^{2}}+\big{[}\mathrm{c}-(\mathrm{a}+\mathrm{b}+1)z\big{]}\frac{dZ(z)}{dz}-\mathrm{a}\mathrm{b}\;Z(z)=0,\end{array}$
(4)

which has only three singular points. In fact, there are several cases in
which the Heun equation is reducible to the hypergeometric equation [7, 8];
some of theses are written in Appendix B because they are relevant for this
article.

We select the form (2) because it is the one used for studying the
substitutions of variables which keep invariant the form of the Heun equation
[5]. These substitutions result from 8 homotopic transformations (no change of
the independent variable) and 24 linear fractional transformations of the
independent variable. As in the case of the hypergeometric equation, the
substitutions can also be used to transform an initial solution into new
solutions which, for the Darboux equation, can be rewritten in terms of
elliptic functions. Thus, from an initial solution, the homotopic
transformations generate a group containing 8 expansions. New groups can be
generated by firstly applying on the initial solution a fractional
transformation of the independent variable.

As initial solutions for Heun’s equation we take a familiar power series
around the origin of the independent variable [4, 9] and an expansion in
series of Gauss hypergeometric functions. As a matter of fact, in Section 3 we
consider two groups of power series whose initial solutions are the expansions
having the form

$\displaystyle\mathring{H}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\mathring{b}_{n}^{(1)}x^{n}\qquad\mbox{and}\qquad\mathring{\bm{H}}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\mathring{\bm{b}}_{n}^{(1)}(1-x)^{n},$
(5)

around the points $x=0$ and $x=1$. We will find that $\mathring{\bm{H}}^{(1)}$
results from $\mathring{H}^{(1)}$ by means of the linear fractional
transformation $x\mapsto 1-x$. Furthermore, the homotopic transformations lead
to 16 expansions, $\mathring{H}^{(i)}$ and $\mathring{\bm{H}}^{(i)}$
($i=1,2,\cdots,8$). The mathring symbol $\mathring{}$ over
$\mathring{H}^{(i)}$ and $\mathring{\bm{H}}^{(i)}$ distinguishes these
expansions from the expansions in series of hypergeometric functions,
${H}^{(i)}$ and ${\bm{H}}^{(i)}$.

On the other side, in Section 4 we consider two groups of expansions in series
of hypergeometric functions, $F(\mbox{a},\mbox{b};\mbox{c};z)$, generated from

$\displaystyle{{H}}^{(1)}(x)=\begin{array}[]{l}\left(1-\frac{x}{a}\right)^{-\alpha}\end{array}\displaystyle\sum_{n=0}^{\infty}\begin{array}[]{l}\frac{b_{n}^{(1)}}{\Gamma(n+\gamma)}\left[\frac{(a-1)x}{a-x}\right]^{n}F\left[n+\alpha,\gamma+\delta-\alpha-1;n+\gamma;\frac{(a-1)x}{a-x}\right],\end{array}$
(8)
$\displaystyle{\bm{H}}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\begin{array}[]{l}\frac{\bm{b}_{n}^{(1)}}{\Gamma(n+\delta)}(1-x)^{n}F\left(n+\alpha,\gamma+\delta-\alpha-1;n+\delta;1-x\right).\end{array}$
(10)

The expansions (8) and (10) will be obtained from an expansion having the form

$\displaystyle\begin{array}[]{l}\bar{H}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\bar{b}_{n}^{(1)}x^{n}F\left(n+\alpha,\gamma+\delta-\alpha-1;n+\gamma;x\right).\end{array}$
(12)

In effect, we will find that the linear fractional substitutions
$x\mapsto{(a-1)x}/{(a-x)}$ and $x\mapsto 1-x$ transform $\bar{H}^{(1)}$ into
${{H}}^{(1)}$ and ${\bm{H}}^{(1)}$, respectively. The homotopic
transformations lead once again to 16 expansions, ${{H}}^{(i)}$ and
${{H}}^{(i)}$. The arguments $z$ of the above
$F(\mbox{a},\mbox{b};\mbox{c};z)$ are appropriate for the Darboux equation
because for this case $|z|\leq 1$, as we will see. For some constraints
between the parameters, the hypergeometric-function expansions reduce to
power-series expansions; for example, $\bar{H}^{(1)}=\mathring{H}^{(1)}$ if
$\gamma+\delta=\alpha+1$. We will see as well that the initial solutions,
$\mathring{H}(x)$ and $\bar{H}^{(1)}(x)$, lead to known solutions for the
confluent Heun equation.

The power-series and hypergeometric-function expansions are both given by one-
sided infinite series ($n\geq 0$) whose coefficients obey three-term
recurrence relations. These relations constitute a system of homogeneous
linear equations which admits non trivial solutions only if the determinant of
the corresponding matrix vanish (characteristic equation). This fact requires
a free parameter in the differential equation. The absence of free parameters
would demand two-sided infinite series ($-\infty<n<+\infty$) having a
parameter which is not present in the differential equation, as in the case of
Mathieu equation (see chapter VI of Ref. [3]).

Under certain conditions, the one-sided series truncate on the right at $n=N$,
becoming finite series with $0\leq n\leq N$. For finite series the convergence
is decided by examining each term of the series, provided that the
characteristic equation is satisfied. For infinite series the convergence
requires some ratio test between successive terms of the series. In the
present article, only the power expansions $\mathring{H}^{(i)}$ are intended
to be used as infinite-series solutions; by this reason, there is no concern
about ratio tests for the other expansions.

Finite-series solutions occur in quasi-exactly solvable (QES) problems of
quantum mechanics [10, 11], for which one part of the energy spectrum and the
corresponding eigenfunctions can be computed explicitly. In fact, a QES
problem is sometimes defined as a problem which presents solutions given by
finite series with coefficients satisfying three-term or higher order
recurrence relations [12]. The “quasi-exactly solvable” part of the spectrum
is obtained from the finite-series solutions, but there are also infinite-
series solutions which are convergent and bounded, as we will see.

In Section 2 we explain how to get solutions for the Darboux equation from
solutions of the Heun equation and show that, for some elliptic potentials,
the Schrödinger equation reduces to the Darboux equation. In addition, we
collect mathematical informations used in subsequent sections: properties of
the hypergeometric functions, transformations of the Heun equation and some
aspects of the three-term recurrence relations for the series coefficients.

In Section 3 we present the power series solutions $\mathring{H}^{(i)}$ and
$\mathring{\bm{H}}^{(i)}$, while in Section 4 we construct the expansions
${H}^{(i)}$ and $\bm{H}^{(i)}$ in series in hypergeometric functions. The
solutions of the four groups are generated straightforward by applying
substitutions of variable on the expansion $\mathring{H}^{(1)}(x)$ written in
(5), and on $\bar{H}^{(1)}(x)$ written in (12). Despite this, all the
expansions are written explicitly for the purposes of applications.

The largest part of the present study is Section 5 which deals with a family
of QES potentials which appear in band theory of solids. In this case, the
equation reduces to an associated Lamé equation and admits finite-series
solutions resulting from power series expansions and from expansions in series
of hypergeometric functions. In addition to finite-series eigenfunctions
(which characterize the quasi-exact solvability) we get as well infinite-
series eigenfunctions, convergent and bounded for any value of the independent
variable.

Final remarks and additional references are in Section 6, Appendix A gives the
homotopic transformations of the Heun equation, while in Appendix B the Heun
equation is reduced to hypergeometric equation when its parameters satisfy
certain constraints. In Appendix C, we write degenerate solutions of the
Schrödinger equation for some particular values of the parameters of the
associated Lamé potential. Eventually, in Appendix D we discuss the
possibility of developing new solutions for the Darboux equation by starting
from expansions in series of hypergeometric functions (for the Heun equation)
given by Svartholm [13] and by Erdélyi [14, 15].

##  2 Mathematical Preliminaries

In this section the Darboux equation is written as a particular case of the
Heun equation, while the Schrödinger equation is written as a Darboux equation
for four known elliptic potentials. We write also some properties of the Gauss
hypergeometric functions and select the transformations of variables of the
Heun equation which will be used to generate solutions appropriate for the
Darboux equation. Finally we regard three-term recurrence relations for the
series coefficients and state a known theorem which gives conditions
sufficient for assuring real and distinct solutions for the characteristic
equation of finite series.

###  2.1 Elliptic Darboux equation and the Schrödinger equation

The basic Jacobian elliptic functions $\mathrm{sn}(u,k)$, $\mathrm{cn}(u,k)$
and $\operatorname{dn}(u,k)$ are written as $\operatorname{sn}{u}$,
$\operatorname{cn}{u}$ and $\operatorname{dn}{u}$, respectively [3, 16]. Here
the parameter $k$, called modulus of the functions, is supposed to be a real
number such that $0<k^{2}<1$. These functions satisfy the relations

${\rm sn}^{2}u+{\rm cn}^{2}u=k^{2}{\rm sn}^{2}u+{\rm
dn}^{2}u=1\;\Rightarrow\;(k^{2}-1){\rm sn}^{2}u={\rm cn}^{2}u-{\rm
dn}^{2}u,\;1-k^{2}={\rm dn}^{2}u-k^{2}{\rm cn}^{2}u.$ (13)

Beside these, we will use the functions $\operatorname{cd}u$ and
$\operatorname{sd}u$, defined by

$\displaystyle\operatorname{cd}u={\operatorname{cn}u}/{\operatorname{dn}u},\qquad\operatorname{sd}u={\operatorname{sn}u}/{\operatorname{dn}u}.$
(14)

Now we obtain the Darboux equation from the Heun equation (2) by taking
$a=k^{-2}$ and achieving the substitutions

$\displaystyle\begin{array}[]{l}H\left[x(u)\right]=\left({\rm
sn}^{2}u\right)^{\frac{1-2\gamma}{4}}\left({\rm
cn}^{2}u\right)^{\frac{1-2\delta}{4}}\left({\rm
dn}^{2}u\right)^{\frac{1-2\epsilon}{4}}U(u)\end{array},\quad x(u)={\rm
sn}^{2}u,\quad\left[a=k^{-2}\right].$ (16)

which lead to Eq. (1) expressed in terms of the parameters of the Heun
equation, namely,

$\displaystyle\begin{array}[]{l}\frac{d^{2}U(u)}{du^{2}}+\Big{\\{}h-\left[(\alpha-\beta)^{2}-\frac{1}{4}\right]k^{2}{\rm
sn}^{2}u-\left(\gamma-\frac{1}{2}\right)\left(\gamma-\frac{3}{2}\right)\frac{1}{{\rm
sn^{2}}u}-\end{array}\vspace{2mm}$ (18)
$\displaystyle\begin{array}[]{l}\left(\delta-\frac{1}{2}\right)\left(\delta-\frac{3}{2}\right)\frac{{\rm
dn^{2}}u}{{\rm
cn^{2}}u}-\left(\epsilon-\frac{1}{2}\right)\left(\epsilon-\frac{3}{2}\right)\frac{k^{2}\;{\rm
cn}^{2}u}{{\rm dn^{2}}u}\Big{\\}}U(u)=0,\quad[\mbox{Darboux Eq,}]\end{array}$
(20)

where
$h=(\gamma+\delta)^{2}+1-2\gamma-2\delta-\left[4q-(\gamma+\epsilon)^{2}-1+2\gamma+2\epsilon\right]k^{2}$.
Therefore, solutions $U(u)$ for the Darboux equation (18) follow from
solutions $H(x)$ for the Heun equation through (16). The associated Lamé
equation and the Lamé equation occur when, in Eq. (18),

$\displaystyle\begin{array}[]{l}(\gamma,\delta)=\left(\frac{1}{2},\frac{1}{2}\right),\;\left(\frac{1}{2},\frac{3}{2}\right),\;\left(\frac{3}{2},\frac{1}{2}\right),\;\left(\frac{3}{2},\frac{3}{2}\right)\qquad[\text{associated
Lam\'{e} equation}];\end{array}$ (22) $\displaystyle(\gamma,\delta,\epsilon)$
$\displaystyle=$
$\displaystyle\begin{array}[]{l}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right),\;\left(\frac{1}{2},\frac{1}{2},\frac{3}{2}\right),\;\left(\frac{1}{2},\frac{3}{2},\frac{1}{2}\right),\;\left(\frac{1}{2},\frac{3}{2},\frac{3}{2}\right),\end{array}$
(26)
$\displaystyle\begin{array}[]{l}\left(\frac{3}{2},\frac{1}{2},\frac{1}{2}\right),\;\left(\frac{3}{2},\frac{1}{2},\frac{3}{2}\right),\;\left(\frac{3}{2},\frac{3}{2},\frac{1}{2}\right),\;\left(\frac{3}{2},\frac{3}{2},\frac{3}{2}\right)\qquad[\text{Lam\'{e}
equation}].\end{array}$

Next we regard the one-dimensional stationary Schrödinger equation,

$\displaystyle\begin{array}[]{l}\frac{d^{2}\psi(u)}{du^{2}}+\big{[}{\cal
E}-{\cal V}(u)\big{]}\psi(u)=0,\quad u=\kappa\mbox{x},\ \quad{\cal
E}=\frac{2M}{\hbar^{2}\kappa^{2}}E,\quad{\cal
V}(u)=\frac{2M}{\hbar^{2}\kappa^{2}}{V}(\mbox{x}),\end{array}$ (28)

for a particle with mass $M$ and energy $E$ – the constant $\kappa$ has
inverse-length dimension, $\hbar$ is the Plank constant divided by $2\pi$, x
is the spatial coordinate and ${V}(\mbox{x})$ is the potential. Eq. (28)
reduces to the Darboux equation (18), with $\psi(u)=U(u)$, for the following
potentials ${\cal V}_{i}(u)$ ($i=1,2,3,4$). The three Ganguly’s potentials
[17, 18]:

$\displaystyle\begin{array}[]{l}{\cal
V}_{1}(u)=(1-k^{2})\left[\frac{2}{\mathrm{cn}^{2}u}-\frac{(\bm{l}+2)(\bm{l}+3)}{\mathrm{dn}^{2}u}\right]\end{array}\vspace{2mm}$
(30) $\displaystyle\hskip
28.45274pt\begin{array}[]{l}\stackrel{{\scriptstyle\text{Eq.(\ref{imply})}}}{{=\joinrel=\joinrel=}}-2k^{2}-(\bm{l}+2)(\bm{l}+3)+2\;\frac{\mathrm{dn}^{2}u}{\mathrm{cn}^{2}u}+(\bm{l}+2)(\bm{l}+3)\frac{k^{2}\;\mathrm{cn}^{2}u}{\mathrm{dn}^{2}u}.\end{array}\vspace{3mm}$
(32) $\displaystyle\begin{array}[]{l}{\cal
V}_{2}(u)=\frac{2}{\mathrm{sn}^{2}u}-\frac{(1-k^{2})(\bm{l}+2)(\bm{l}+3)}{\mathrm{dn}^{2}u}\end{array}\begin{array}[]{l}\stackrel{{\scriptstyle\text{(\ref{imply})}}}{{=}}-(\bm{l}+2)(\bm{l}+3)+\frac{2}{\mathrm{sn}^{2}u}+(\bm{l}+2)(\bm{l}+3)\frac{k^{2}\;\mathrm{cn}^{2}u}{\mathrm{dn}^{2}u}.\end{array}\vspace{2mm}$
(35) $\displaystyle\begin{array}[]{l}{\cal
V}_{3}(u)=\bm{m}(\bm{m}+1)\;k^{2}\mathrm{sn}^{2}u+\bm{l}(\bm{l}+1)\frac{k^{2}\;{\rm
cn}^{2}u}{{\rm dn^{2}}u},\end{array}$ (37)

where $\bm{l}$ and $\bm{m}$ are constants. For the potential ${\cal V}_{3}(u)$
the Schrödinger equation becomes an associated Lamé equation which was firstly
considered by Khare and Sukhatme [19, 20]. On the other side, there is the
Ushveridze potential [11]

$\displaystyle\begin{array}[]{l}{\cal
V}_{4}(u)=-4\left(\bm{a}-\frac{1}{4}\right)\left(\bm{a}-\frac{3}{4}\right){\rm
dn}^{2}u+4\left(\bm{b}-\frac{1}{4}\right)\left(\bm{b}-\frac{3}{4}\right)\frac{{\rm
dn^{2}u}}{{\rm
sn^{2}u}}+4\left(\bm{c}-\frac{1}{4}\right)\left(\bm{c}-\frac{3}{4}\right)\frac{{\rm
dn^{2}u}}{{\rm cn^{2}u}}\end{array}\vspace{3mm}$ $\displaystyle\hskip
45.52458pt+\begin{array}[]{l}4\left(\bm{a}+\bm{b}+\bm{c}+\bm{l}-\frac{1}{4}\right)\left(\bm{a}+\bm{b}+\bm{c}+\bm{l}-\frac{3}{4}\right)k^{2}(k^{2}-1)\frac{{\rm
sn^{2}u}}{{\rm dn^{2}u}},\end{array}$

where $\bm{a}$, $\bm{b}$, $\bm{c}$ and $\bm{l}$ are real constants. By
rewriting ${\cal V}_{4}$ as

$\displaystyle\begin{array}[]{l}{\cal
V}_{4}(u)\stackrel{{\scriptstyle\text{(\ref{imply})}}}{{=}}-4\left(\bm{a}-\frac{1}{4}\right)\left(\bm{a}-\frac{3}{4}\right)-4\left(\bm{a}+\bm{c}+\bm{l}\right)\left(\bm{a}+2\bm{b}+\bm{c}+\bm{l}-1\right)k^{2}\end{array}$
(41) $\displaystyle\hskip
8.5359pt\begin{array}[]{l}-8\left(\bm{b}-\frac{1}{4}\right)\left(\bm{b}-\frac{3}{4}\right)k^{2}+4\left(\bm{a}-\frac{1}{4}\right)\left(\bm{a}-\frac{3}{4}\right)k^{2}{\rm
sn}^{2}u+4\left(\bm{b}-\frac{1}{4}\right)\left(\bm{b}-\frac{3}{4}\right)\frac{1}{{\rm
sn^{2}u}}\end{array}$ (43) $\displaystyle\hskip
8.5359pt\begin{array}[]{l}+4\left(\bm{c}-\frac{1}{4}\right)\left(\bm{c}-\frac{3}{4}\right)\frac{{\rm
dn^{2}u}}{{\rm
cn^{2}u}}+4\left(\bm{a}+\bm{b}+\bm{c}+\bm{l}-\frac{1}{4}\right)\left(\bm{a}+\bm{b}+\bm{c}+\bm{l}-\frac{3}{4}\right)\frac{k^{2}\;{\rm
cn}^{2}u}{{\rm dn^{2}u}},\end{array}\qquad$ (45)

the Schrödinger equation (28) once more takes the form (18) for the Darboux
equation.

Thus, if we know solutions $H(x)$ for the Heun equation, the eigenfunctions
$\psi(u)$ follow from

$\displaystyle\psi(u)=U(u)\stackrel{{\scriptstyle\text{(\ref{substituicoes})}}}{{=}}\left({\rm
sn}{u}\right)^{\frac{2\gamma-1}{2}}\left({\rm
cn}{u}\right)^{\frac{2\delta-1}{2}}\left({\rm
dn}{u}\right)^{\frac{2\epsilon-1}{2}}H\left(x\right),\qquad\left[a=k^{-2},\;x={\rm
sn}^{2}u\right],$ (46)

provided that these $\psi(u)$ are bounded for any admissible value of the
variable $u$. The problem is QES if there are solutions $H(x)$ given by finite
series, a fact which occurs when the parameters of the Schrödinger equation
satisfy certain conditions. For the associated Lamé potential (37) the
conditions stated in [20] are: (i) either $\bm{m}-\bm{l}\neq 0$ or
$\bm{l}+\bm{m}$ is an integer, (ii) either $\bm{l}$ or $\bm{m}$ is an integer
and the other is half an odd integer. In addition, the second case admits
doubly degenerate eigenfunctions, that is, two independent eigenfunctions with
the same value for the energy.

###  2.2 The Gauss hypergeometric function

The hypergeometric function $F(\mathrm{a,b;c};z)$, solution of Eq. (4), is
defined by the series [6]

$\displaystyle
F\left(\mathrm{a,b;c};z\right)=F(\mathrm{b,a;c};z)=1+\frac{\mathrm{ab}}{1!\
\mathrm{c}}z+\frac{\mathrm{a(a+1)b(b+1)}}{2!\
\mathrm{c(c+1)}}z^{2}+\cdots=\sum_{n=0}^{\infty}\frac{(\mathrm{a})_{n}(\mathrm{b})_{n}}{n!\;(\mathrm{c})_{n}}z^{n},$

where $(\mathrm{a})_{n}$, $(\mathrm{b})_{n}$ and $(\mathrm{c})_{n}$ are
Pochhammer symbols: $(\mathrm{a})_{0}=1$, $\cdots$,
$(\mathrm{a})_{n}=\mathrm{a}(\mathrm{a}+1)\cdots(\mathrm{a}+n-1)$ $=$
$\Gamma(\mathrm{a}+n)/\Gamma(\mathrm{a})$, etc. The above series is not
defined when the parameter $\mathrm{c}$ is zero or negative integer. Besides
this, if the parameter $\mathrm{a}$ or the parameter $\mathrm{b}$ is zero or
negative integer, the series reduces to a polynomial. A infinite series
$F\left(\mathrm{a,b;c};z\right)$ converges absolutely for $|z|<1$ and diverges
for $|z|>1$; by the Raabe test, the series converges on the circle $|z|=1$ if
Re$(c-a-b)>0$. Thus,

$\displaystyle F\left(\mathrm{a,b;c};z\right)$ $\displaystyle\mbox{converges
absolutely for }|z|<1,\mbox{ diverges for }|z|>1,$ (47)
$\displaystyle\mbox{converges also on }|z|=1\mbox{ if Re(c-a-b)}>0.$

The following relations will be useful:

$\displaystyle\begin{array}[]{l}F(\mathrm{a,b;c};z)=(1-z)^{\mathrm{c-a-b}}F(\mathrm{c-a,c-b;c;z})\quad\mbox{for}\quad|z|<1,\end{array}\vspace{2mm}$
(49)
$\displaystyle\begin{array}[]{l}F\left(\mathrm{a,b;c};1\right)=\frac{\Gamma(\mathrm{c})\Gamma(\mathrm{c-a-b})}{\Gamma(\mathrm{c-a})\Gamma(\mathrm{c-b})}\quad\mbox{if}\quad\mbox{Re(c-a-b)}>0.\end{array}$
(51)

In Section 4 we also use the relations [6]

$\displaystyle\begin{array}[]{l}\frac{d}{dz}\big{[}z^{\mathrm{c}-1}(1-z)^{\mathrm{b}-\mathrm{c}+1}F(\mathrm{a},\mathrm{b};\mathrm{c};z)\big{]}=(\mathrm{c}-1)z^{\mathrm{c}-2}(1-z)^{\mathrm{b}-\mathrm{c}}F(\mathrm{a}-1,\mathrm{b};\mathrm{c}-1;z),\vspace{3mm}\\\
\frac{d}{dz}\big{[}(1-z)^{\mathrm{a}}F(\mathrm{a},\mathrm{b};\mathrm{c};z)\big{]}=-\frac{\mathrm{a}(\mathrm{c}-\mathrm{b})}{\mathrm{c}}(1-z)^{\mathrm{a}-1}F(\mathrm{a}+1,\mathrm{b};\mathrm{c}+1;z),\end{array}$
(54)

which imply

$\displaystyle\begin{array}[]{l}(1-z)\frac{d}{dz}F(\mathrm{a,b;c};z)=\left(\mathrm{b}-\frac{\mathrm{c}-1}{z}\right)F(\mathrm{a,b;c};z)+\frac{\mathrm{c}-1}{z}F(\mathrm{a-1,b;c-1};z),\vspace{3mm}\\\
(z-1)\frac{d}{dz}F(\mathrm{a,b;c};z)=-\mathrm{a}F(\mathrm{a,b;c};z)+\frac{\mathrm{a}(\mathrm{c}-\mathrm{b})}{\mathrm{c}}F(\mathrm{a+1,b;c+1};z).\end{array}$
(57)

###  2.3 Some transformations of Heun’s general equation

In 2007 Maier set down the 192 substitutions of variables which preserve the
form of Heun’s equation. The table 2 of his paper [5] displays the
transformations of each parameter and variable of the equation. These
transformations follow from the composition of 8 homotopic substitutions of
the dependent variable $H(x)$ (which keep unaltered the independent variable
$x$) with 24 linear fractional substitutions $x\mapsto y(x)=(Ax+B)/(Cx+D)$ (in
general these demand a transformation of the independent variable as well). We
rewrite some of these substitutions in a form convenient for applications.

We denote by $H(x)=H(q,a;\alpha,\beta,\gamma,\delta;x)$ a solution of the Heun
equation (2); on the other hand, the transformations of $H(x)$ are indicated
by the symbol $M_{i}$ following the order in which they appear in Maier´s
table $(i=1,2,\cdots,192)$. Hence, each $M_{i}$ transform $H(x)$ into
$M_{i}H(x)$ according to

$\displaystyle
M_{i}H(x)=M_{i}H(a,q;\alpha,\beta,\gamma,\delta;x)=f_{i}(x)H\left[\tilde{a},\tilde{q};\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta};y(x)\right].$
(58)

where $f_{i}(x)$ and
$\left[\tilde{q},\tilde{a};\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta};y(x)\right]$
depend on the transformation considered. For example, the fractional
substitutions

$\displaystyle\begin{array}[]{l}\frac{(1-a)x}{x-a},\qquad
1-x,\qquad\frac{a(x-1)}{x-a}\end{array}$ (60)

may be represented by $M_{17}$, $M_{49}$ and $M_{65}$, that is,

$\displaystyle\begin{array}[]{l}M_{17}H(x)=\left(1-\frac{x}{a}\right)^{-\alpha}H\left[1-a,-q+\alpha\gamma;\alpha,-\beta+\gamma+\delta,\gamma,\delta;\frac{(1-a)x}{x-a}\right],\end{array}\vspace{2mm}$
(62)
$\displaystyle\begin{array}[]{l}M_{49}H(x)=H(1-a,-q+\alpha\beta;\alpha,\beta,\delta,\gamma;1-x),\end{array}\vspace{2mm}$
(64)
$\displaystyle\begin{array}[]{l}M_{65}H(x)=\left(1-\frac{x}{a}\right)^{-\alpha}H\left[a,q-\alpha(\beta-\delta);\alpha,-\beta+\gamma+\delta,\delta,\gamma;\frac{a(x-1)}{x-a}\right].\end{array}$
(66)

As a matter of fact, for each substitution of the independent variable, there
are 8 substitutions of the dependent variable on account of the homotopic
transformations. These are given by the transformations $M_{1}-M_{4}$ and
${M}_{25}-M_{28}$ of Maier’s table but here, as in [21] and [22], they are
denoted by $T_{i}$ ($i=1,\cdots,8$) according to

$\displaystyle T_{1}=M_{1},\quad T_{2}=M_{25},\quad T_{3}=M_{2},\quad
T_{4}=M_{26},\vspace{2mm}$ $\displaystyle T_{5}=M_{3},\quad T_{6}=M_{27},\quad
T_{7}=M_{4},\quad T_{8}=M_{28}.$ (67)

The 8 transformations are written in Appendix A.

Notice that for the Darboux equation (18)

$\displaystyle\begin{array}[]{l}x=\mathrm{sn}^{2}u,\qquad
1-x=\mathrm{cn}^{2}u,\qquad\frac{(a-1)x}{a-x}=(1-k^{2})\;\mathrm{sd}^{2}u,\qquad\frac{a(x-1)}{x-a}=\mathrm{cd}^{2}u,\end{array}$
(69)

where $\operatorname{sd}u$ and $\operatorname{cd}u$ are the functions given in
(14). From definitions [3, 16], we know that $0\leq\mathrm{sn}^{2}u\leq 1$,
$0\leq\mathrm{cn}^{2}u\leq 1$ and $0<\mathrm{dn}^{2}u<1$. Further, the
relations [3]

$\displaystyle\begin{array}[]{l}\mathrm{cd}\;u=\mathrm{sn}(u+K)\quad\mbox{and}\quad(1-k^{2})\;\mathrm{sd}^{2}u=1-\mathrm{cd}^{2}u,\end{array}\vspace{3mm}$
(71) $\displaystyle\mbox{where
}K=K(k)=\int_{0}^{\frac{\pi}{2}}\left[1-k^{2}\sin^{2}\theta\right]^{-\frac{1}{2}}d\theta,$

assure that $0\leq\mathrm{cd}^{2}u\leq 1$ and
$0\leq(1-k^{2})\;\mathrm{sd}^{2}u\leq 1$. Thus, the variables (69) are inside
the interval $[0,1]$ which is suitable for arguments of the hypergeometric
functions, according to (47). Similarly, the variables $x/a$ and $1-(x/a)$ are
also appropriate. As we are interested in Darboux equation, in sections III
and IV we shall use the homotopic transformations and the transformations
$M_{17}$ and $M_{49}$ given in Eqs. (62) and (64).

Therefore, in most cases we will use the substitutions of variables as a
procedure to generate new solutions out of a given solution $H(x)$. However,
the substitutions can be used to simply transform the Heun equation (2) into
another version of itself. We will use this viewpoint in Appendix B in order
to show that, for certain values of the parameters, the Heun equation (2) is
transformed into an equation reducible to the hypergeometric equation (4).

###  2.4 Three-term recurrence relations and convergence

The following are known results which are also valid for the confluent cases
of the Heun equation [21]. The three-term recurrence relations for the
coefficients $b_{n}$ of one-sided solutions have the form

$\displaystyle\alpha_{0}b_{1}+\beta_{0}b_{0}=0,\qquad\alpha_{n}b_{n+1}+\beta_{n}b_{n}+\gamma_{n}b_{n-1}=0\qquad(n\geq
1),$ (72)

where $\alpha_{n}$, $\beta_{n}$ and $\gamma_{n}$ are functions of the
parameters of the equation. This system of homogeneous linear equations can be
written in matrix form. Then, we see that the system has non-trivial solutions
only if the determinant of a tridiagonal coefficient matrix vanishes; such
condition is called ‘characteristic equation’ and determine the possible
values of some ‘free’ parameter of the equation. The characteristic equation
is also written as the continued fraction [23, 24]

$\displaystyle\beta_{0}=\frac{\alpha_{0}\gamma_{1}}{\beta_{1}-}\
\frac{\alpha_{1}\gamma_{2}}{\beta_{2}-}\
\frac{\alpha_{2}\gamma_{3}}{\beta_{3}-}\cdots.$ (73)

A series whose coefficients satisfy three-term recurrence relations like (72)
truncates on the right-hand side if $\gamma_{n}=0$ for some value of $n\geq$.
Precisely, if $\gamma_{N+1}=0$, the series becomes a finite series with $0\leq
n\leq N$ [25]). So, we can write

$\displaystyle\begin{array}[]{l}\gamma_{N+1}=0\quad\Rightarrow\quad\text{finite
series with }n\leq N.\end{array}$ (75)

For finite series, the characteristic equation is equivalent to the vanishing
of the determinant of a $(N+1)\times(N+1)$ matrix. Precisely

$\displaystyle\det\left[\begin{array}[]{ccccccccc}\beta_{0}&\alpha_{0}&0&&\\\
\gamma_{1}&\beta_{1}&\alpha_{1}&\\\ &\ddots&\ddots&\ddots&\\\
&&\gamma_{N-1}&\beta_{N-1}&\alpha_{N-1}\\\
&&0&\gamma_{N}&\beta_{N}\end{array}\right]=0$ (81)

assures that there are nontrivial solutions for the coefficients $b_{n}$
($n=0,\;1,\cdots,N$).

The following theorem, due to Arscott [25, 26], holds for finite series:

Let $\beta_{i}=\mathcal{B}_{i}-\Lambda$ for $i=0,1,\cdots,N$, where $\Lambda$
is an arbitrary constant. If $\alpha_{i}$, $\mathcal{B}_{i}$ and $\gamma_{i}$
are all real and independent of $\Lambda$, and if each product
$\alpha_{i-1}\;\gamma_{i}$ is positive, that is,

$\mbox{\it if }\alpha_{i},\;\mathcal{B}_{i},\;\gamma_{i}\mbox{ \it are real
and independent of }\Lambda,\;\mbox{\it and if
}\;\alpha_{i-1}\;\gamma_{i}>0\;\mbox{\it for}\;i=1,\cdots,N,\quad$ (82)

then the vanishing of the determinant of the above matrix affords $N+1$ real
and distinct values for $\Lambda$; if $\gamma_{i}\;\alpha_{i-1}\leq 0$ nothing
can be said about the nature of $\Lambda$.

The previous theorem will be useful to analyse the energy spectra of finite-
series wavefunctions of quasi-exactly solvable problems, as the one discussed
in Section 5. In these cases the characteristic equation becomes an algebraic
equation of order $N+1$ in $\Lambda$, where $\Lambda$ is a linear function of
the energy ${\cal E}$. For each value of ${\cal E}$, the series coefficients
can be computed by a forward application of the recurrence relations, from
$b_{0}$ upward to $b_{N}$; this gives all the $b_{n}$ in terms of $b_{0}$. In
this case the convergence is decided by examining each term of the series.

On the other side, the convergence of infinite-series solutions comes from
ratio tests for successive terms of the series [27, 28]. In general the three-
term relations (72) present two linearly independent solutions [9, 24] and,
consequently, two ratios for $b_{n+1}/b_{n}$ but, in order to assure the
convergence the continued fraction (73), we have to choose the solution that
Gautschi has called minimal (or minimal at infinity): if $f_{n}$ and $g_{n}$
denote such solutions and $\lim_{n\to\infty}(f_{n}/g_{n})=0$, then $f_{n}$ is
the minimal solution. Alternatives to the term minimal are give in a footnote
on page 25 of Gautschi’s paper [24]. For practical purposes, if

$\displaystyle\lim_{n\to\infty}\left({b_{n+1}}/{b_{n}}\right)=t_{1},\qquad\lim_{n\to\infty}\left({b_{n+1}}/{b_{n}}\right)=t_{2},\qquad
t_{1}\neq t_{2},$ (83)

then a Poincaré-Perron theorem [24] assures the existence of two linearly
independent solutions presenting these limits, the minimal solution being the
one with smaller modulus for the limit, $|t_{1}|$ or $|t_{2}|$.

The use of a minimal solution demands a backward procedure to compute the
ratios $b_{n+1}/b_{n}$, from a large value of $n$ downward to $n=0$. However,
before this, it is necessary to solve the characteristic equation which
affords relations between the parameters of the Heun equation. Some
considerations concerning solutions of characteristic equation for infinite
series are given in [29, 30, 31]. We just mention that solving Eq. (73) is an
important part of the problem since that equation determines, for example, the
energy spectra of hydrogen molecular ion (confluent Heun equation).

##  3 Power-series solutions

This section regards the two groups of power series solutions,
$\mathring{H}^{(i)}(x)$ around $x=0$ and $\mathring{\bm{H}}^{(i)}(x)$ around
$x=1$ ($i=1,\cdots 8$), each group being closed with respect to the homotopic
transformations (A.2-A.20). The series coefficients are denoted by
$\mathring{b}_{n}^{(i)}$ and $\mathring{\bm{b}}_{n}^{(i)}$, respectively.

The starting-point solution $\mathring{H}^{(1)}(x)$ generates the others by
means of variable transformations as follows

$\displaystyle\begin{array}[]{l}\mathring{H}^{(i)}(x)=T_{i}\mathring{H}^{(1)}(x),\qquad\mathring{\bm{H}}^{(1)}(x)=M_{49}\mathring{H}^{(1)}(x),\qquad\mathring{\bm{H}}^{(i)}(x)=T_{i}\mathring{\bm{H}}^{(1)}(x).\end{array}$
(85)

Thus, the present section uses only the Möbius substitution $M_{49}$
($x\mapsto 1-x$) written in (64). In order to determine the convergence of
$\mathring{H}^{(i)}$ for infinite series it is necessary to suppose that
$|a|\neq 1$. It will be seen as well that (for a fixed value of $i$) both
series $\mathring{H}^{(i)}$ and $\mathring{\bm{H}}^{(i)}$ reduce to finite
series under the same circumstances since
$\mathring{\gamma}_{n}^{(i)}=\mathring{\bm{\gamma}}_{n}^{(i)}$ in condition
(75); however, one of the series may satisfy the Arscott condition (82) while
the other may not satisfy.

All solutions $\mathring{H}^{(i)}$ and $\mathring{\bm{H}}^{(i)}$ will be used
in Section 5. The convergence of infinite series $\mathring{H}^{(i)}$ holds
for minimal solutions of the recurrence relations for the coefficients
$\mathring{b}_{n}^{(i)}$. At the end of Section 3.1, we find conditions for
linear dependence between the expansions $\mathring{H}^{(i)}$ and
$\mathring{H}^{(i+4)}$ for $i=1,2,3,4$. We show as well that the expansion
$\mathring{H}^{(1)}$, by means of a confluence procedure, leads to a known
expansion for solutions of the confluent Heun equation (CHE).

###  3.1 Series $\mathring{H}^{(i)}(x)$ around $x=0$

By using the homotopic transformations $T_{i}$, the solutions around $x=0$ are
written as

$\displaystyle\mathring{H}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\mathring{b}_{n}^{(1)}x^{n},\qquad\mathring{H}^{(i)}(x)=T_{i}\mathring{H}^{(1)}(x).$
(86)

The coefficients of the recurrence relations (72) for $\mathring{b}_{n}^{(1)}$
are [21]

$\displaystyle\begin{array}[]{l}\mathring{\alpha}_{n}^{(1)}=a(n+1)(n+\gamma),\quad\mathring{\beta}_{n}^{(1)}=-an(n+\gamma+\delta-1)-n(n+\alpha+\beta-\delta)-q,\end{array}\vspace{2mm}$
(88)
$\displaystyle\begin{array}[]{l}\mathring{\gamma}_{n}^{(1)}=(n+\alpha-1)(n+\beta-1).\end{array}$
(90)

Dividing these relations by $n^{2}\mathring{b}_{n}^{(1)}$, we get

$\displaystyle\begin{array}[]{l}a\left[1+\frac{1+\gamma}{n}+O\left(\frac{1}{n^{2}}\right)\right]\frac{\mathring{b}_{n+1}^{(1)}}{\mathring{b}_{n}^{(1)}}-\left[a+1+\frac{a(\gamma+\delta-1)+\alpha+\beta-\delta}{n}+O\left(\frac{1}{n^{2}}\right)\right]+\end{array}\vspace{2mm}$
$\displaystyle\begin{array}[]{l}\left[1+\frac{\alpha+\beta-2}{n}+O\left(\frac{1}{n^{2}}\right)\right]\frac{\mathring{b}_{n-1}^{(1)}}{\mathring{b}_{n}^{(1)}}=0.\end{array}$

Then, the ratios $\mathring{b}_{n+1}^{(1)}/\mathring{b}_{n}^{(1)}$ for large
$n$ are

$\displaystyle\begin{array}[]{l}\frac{\mathring{b}_{n+1}^{(1)}}{\mathring{b}_{n}^{(1)}}\sim\frac{1}{a}\left[1+\frac{\epsilon-2}{n}\right]\;\Leftrightarrow\;\frac{\mathring{b}_{n-1}^{(1)}}{\mathring{b}_{n}^{(1)}}\sim
a\left[1-\frac{\epsilon-2}{n}\right]\quad\text{and}\quad\frac{\mathring{b}_{n+1}^{(1)}}{\mathring{b}_{n}^{(1)}}\sim
1+\frac{\delta-2}{n}\;\Leftrightarrow\;\frac{\mathring{b}_{n-1}^{(1)}}{\mathring{b}_{n}^{(1)}}\sim
1-\frac{\delta-2}{n}.\end{array}$

Thus,
$\displaystyle\lim_{n\to\infty}\mathring{b}_{n+1}^{(1)}\big{/}\mathring{b}_{n}^{(1)}$
is $\frac{1}{a}$ or $1$. If $|a|\neq 1$, the two limits are distinct and by
the Poincaré-Perron theorem, mentioned after Eq. (83),

$\displaystyle\begin{array}[]{l}\mbox{for minimal
solution:}\;\;\frac{\mathring{b}_{n+1}^{(1)}}{\mathring{b}_{n}^{(1)}}\sim\frac{1}{a}\left[1+\frac{\epsilon-2}{n}\right]\;\text{
if }|a|>1,\quad\frac{\mathring{b}_{n+1}^{(1)}}{\mathring{b}_{n}^{(1)}}\sim
1+\frac{\delta-2}{n}\;\text{ if }|a|<1.\end{array}$ (95)

Thus, for minimal solution, the regions of convergence follow from

$\displaystyle\vline\frac{\mathring{b}_{n+1}^{(1)}x^{n+1}}{\mathring{b}_{n}^{(1)}x^{n}}\vline\sim\begin{cases}|x|\left[1+\frac{1}{n}\text{Re}\left(\delta-2\right)\right],&\mbox{if}\
|a|<1,\vspace{2mm}\\\
\big{|}\frac{x}{a}\big{|}\left[1+\frac{1}{n}\text{Re}\left(\epsilon-2\right)\right],&\mbox{if}\
|a|>1.\end{cases}$ (96)

Then, if $|a|<1$ the solution $\mathring{H}^{\\{1\\}}(x)$ converges inside the
circle $|x|=1$ which contains $x=0$ and $x=a$; by the Raabe’s test [27, 28] it
converges also on $|x|=1$ if $\mbox{Re}\;\delta<1$. On the other hand, if
$|a|>1$ the solution converges inside the circle $|x|=|a|$ which contains
$x=0$ and $x=1$; by the Raabe’s test it converges also on $|x|=|a|$ if
$\mbox{Re}\;\epsilon<1$. In sum,

$\displaystyle\begin{array}[]{c}\mbox{for minimal}\\\
\mbox{solution}\end{array}\begin{cases}\mbox{if
}|a|<1,\;\mathring{H}^{(1)}(x)\mbox{ converges for }|x|<1\mbox{ and on
}|x|=1\mbox{ if }\mbox{Re}\;\delta<1;\vspace{2mm}\\\ \mbox{if
}|a|>1,\;\mathring{H}^{(1)}(x)\mbox{ converges for }|x|<|a|\mbox{ and on
}|x|=|a|\mbox{ if }\mbox{Re}\;\epsilon<1.\end{cases}$

Observe that Eq. (83) and the limits preceding Eqs. (95) imply that the
recurrence relations also admit a non-minimal solution.

The remaining solutions $\mathring{H}^{(i)}(x)$ are generated by the homotopic
transformations $T_{i}$ as indicated in (86). This gives the expansions
written below, where the coefficients $\mathring{\beta}_{n}^{(i)}$ are written
as polynomials of degree 2 in $n$.

$\displaystyle\begin{cases}\mathring{H}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\mathring{b}_{n}^{(1)}x^{n},\\\
\mathring{\alpha}_{n}^{(1)}=a(n+\gamma)(n+1),\quad\mathring{\beta}_{n}^{(1)}=-(a+1)n^{2}-[a(\gamma+\delta-1)+\alpha+\beta-\delta)]n\\\
-q,\qquad\mathring{\gamma}_{n}^{(1)}=(n+\alpha-1)(n+\beta-1).\end{cases}$ (98)
$\displaystyle\begin{cases}\mathring{H}^{(2)}(x)=x^{1-\gamma}\displaystyle\sum_{n=0}^{\infty}\mathring{b}_{n}^{(2)}x^{n},\\\
\mathring{\alpha}_{n}^{(2)}=a(n+2-\gamma)(n+1),\quad\mathring{\beta}_{n}^{(2)}=-(a+1)n^{2}-[a(1-\gamma+\delta)+1+\epsilon-\gamma]n\\\
-q-(\delta
a+\epsilon)(1-\gamma),\qquad\mathring{\gamma}_{n}^{(2)}=(n+\alpha-\gamma)(n+\beta-\gamma).\end{cases}$
(99)
$\displaystyle\begin{cases}\mathring{H}^{(3)}(x)=(1-x)^{1-\delta}\displaystyle\sum_{n=0}^{\infty}\mathring{b}_{n}^{(3)}x^{n},\\\
\mathring{\alpha}_{n}^{(3)}=a(n+\gamma)(n+1),\qquad\mathring{\beta}_{n}^{(3)}=-(a+1)n^{2}-[a(1+\gamma-\delta)+\alpha+\beta-\delta]n\\\
-q-a\gamma(1-\delta),\qquad\mathring{\gamma}_{n}^{(3)}=(n+\alpha-\delta)(n+\beta-\delta).\end{cases}$
(100)
$\displaystyle\begin{cases}\mathring{H}^{(4)}(x)=x^{1-\gamma}(1-x)^{1-\delta}\displaystyle\sum_{n=0}^{\infty}\mathring{b}_{n}^{(4)}x^{n},\\\
\mathring{\alpha}_{n}^{(4)}=a(n+2-\gamma)(n+1),\;\;\mathring{\beta}_{n}^{(4)}=-(a+1)n^{2}-[a(3-\gamma-\delta)+1+\epsilon-\gamma]n\\\
-q-a(2-\gamma-\delta)-\epsilon(1-\gamma),\;\;\mathring{\gamma}_{n}^{(4)}=(n+1+\alpha-\gamma-\delta)(n+1+\beta-\gamma-\delta).\end{cases}$
(101)
$\displaystyle\begin{cases}\mathring{H}^{(5)}(x)=\displaystyle\left[1-\frac{x}{a}\right]^{1-\epsilon}\sum_{n=0}^{\infty}\mathring{b}_{n}^{(5)}x^{n},\\\
\mathring{\alpha}_{n}^{(5)}=a(n+\gamma)(n+1),\;\;\mathring{\beta}_{n}^{(5)}=-(a+1)n^{2}-[a(\gamma+\delta-1)+2\gamma+\delta-\alpha-\beta]n\\\
-q-\gamma(1-\epsilon),\qquad\mathring{\gamma}_{n}^{(5)}=(n+\gamma+\delta-\alpha-1)(n+\gamma+\delta-\beta-1).\end{cases}$
(102) $\displaystyle\begin{cases}\mathring{H}^{(6)}(x)=\displaystyle
x^{1-\gamma}\left[1-\frac{x}{a}\right]^{1-\epsilon}\sum_{n=0}^{\infty}\mathring{b}_{n}^{(6)}x^{n},\\\
\mathring{\alpha}_{n}^{(6)}=a(n+2-\gamma)(n+1),\quad\mathring{\beta}_{n}^{(6)}=-(a+1)n^{2}-[a(1-\gamma+\delta)-\alpha-\beta+\\\
\delta+2]n-q-\delta
a(1-\gamma)+\alpha+\beta-\delta-1,\quad\mathring{\gamma}_{n}^{(6)}=(n-\alpha+\delta)(n-\beta+\delta).\end{cases}$
(103)
$\displaystyle\begin{cases}\mathring{H}^{(7)}(x)=\displaystyle(1-x)^{1-\delta}\left[1-\frac{x}{a}\right]^{1-\epsilon}\sum_{n=0}^{\infty}\mathring{b}_{n}^{(7)}x^{n},\\\
\mathring{\alpha}_{n}^{(7)}=a(n+\gamma)(n+1),\qquad\mathring{\beta}_{n}^{(7)}=-(a+1)n^{2}-[a(1+\gamma-\delta)-\alpha-\beta+\\\
2\gamma+\delta]n-q-\gamma
a(1-\delta)-\gamma(1-\epsilon),\qquad\mathring{\gamma}_{n}^{(7)}=(n-\alpha+\gamma)(n-\beta+\gamma).\end{cases}$
(104) $\displaystyle\begin{cases}\mathring{H}^{(8)}(x)=\displaystyle
x^{1-\gamma}(1-x)^{1-\delta}\left[1-\frac{x}{a}\right]^{1-\epsilon}\sum_{n=0}^{\infty}\mathring{b}_{n}^{(8)}x^{n},\\\
\mathring{\alpha}_{n}^{(8)}=a(n+2-\gamma)(n+1),\quad\mathring{\beta}_{n}^{(8)}=-(a+1)n^{2}-[a(3-\gamma-\delta)+2-\alpha-\beta\\\
+\delta]n-q-a(2-\gamma-\delta)+\alpha+\beta-\delta-1,\qquad\mathring{\gamma}_{n}^{(8)}=(n+1-\alpha)(n+1-\beta).\end{cases}$
(105)

Since $x$ and $a$ are kept fixed by homotopic transformations $T_{i}$, all the
solutions $\mathring{H}^{(i)}(x)$ are Heun functions which converge inside the
circle $|x|=1$ if $|a|<1$ and inside the circle $|x|=|a|$ if $|a|>1$; in fact,
by

$\displaystyle\begin{array}[]{c}\mbox{by D'Alembetert's}\\\
\mbox{test}\end{array}\begin{cases}\mbox{if }|a|<1\;\mathring{H}^{(i)}\mbox{
converges for }|x|<1\;(\Rightarrow x=0\mbox{ and }x=a);\vspace{2mm}\\\
\mbox{if }|a|>1\;\mathring{H}^{(i)}\mbox{ converges for }|x|<|a|\;(\Rightarrow
x=0\mbox{ and }x=1).\end{cases}$ (108)

In addition, by Raabe’s test, the transformations (A.2-A.20) give the
following conditions for convergence on $|x|=1$ and $|x|=|a|$:

$\displaystyle\mbox{if }|a|<1$
$\displaystyle\begin{cases}\mathring{H}^{(1)},\;\mathring{H}^{(2)},\;\mathring{H}^{(5)}\text{
and }\mathring{H}^{(6)})\text{ converge on }|x|=1\mbox{ if
}\mbox{Re}\;\delta<1;\vspace{2mm}\\\
\mathring{H}^{(3)},\;\mathring{H}^{(4)},\;\mathring{H}^{(7)}\text{ and
}\mathring{H}^{(8)}\text{ converge on }|x|=1\mbox{ if
}\mbox{Re}\;\delta>1;\end{cases}\vspace{2mm}$ (109) $\displaystyle\mbox{if
}|a|>1$
$\displaystyle\begin{cases}\mathring{H}^{(1)},\;\mathring{H}^{(2)},\;\mathring{H}^{(3)}\text{
and }\mathring{H}^{(4)}\text{ converge on }|x|=|a|\mbox{ if
}\mbox{Re}\;\epsilon<1;\vspace{2mm}\\\
\mathring{H}^{(5)},\;\mathring{H}^{(6)},\;\mathring{H}^{(7)}\text{ and
}\mathring{H}^{(8)}\text{ converge on }|x|=|a|\mbox{ if
}\mbox{Re}\;\epsilon>1.\end{cases}$ (110)

The previous restrictions on $\delta$ and $\epsilon$ hold only for infinite
series since the convergence of finite series is determined by examining each
term of the series. Anyway, the behaviour of the solutions at the singular
points depends also on the multiplicative factors which preceed the series.

Since the recurrence relations admit two solutions, Kristensson [9] used the
Raabe test to determine the convergence of $\mathring{H}^{(1)}(x)$ by
supposing that that $|a|>1$. He found that

$\displaystyle\mbox{if }|a|>1\;\mathring{H}^{(1)}\mbox{ converges
for}\left\\{\begin{array}[]{l}|x|<1\mbox{ and at }|x|=1\mbox{ if
}\mbox{Re}\;\delta<1\;(\mbox{dominant solution});\vspace{2mm}\\\ |x|<|a|\mbox{
and at }|x|=|a|\mbox{ if }\mbox{Re}\;\epsilon<1\;(\mbox{minimal
solution}).\end{array}\right.$ (113)

Note the differences between the above treatments: we are considering only
minimal solutions for both $|a|>1$ and $|a|<1$; in addition, we are using the
homotopic transformations.

In the previous expansions we see that

$\displaystyle\begin{array}[]{lllll}a=0,&\Rightarrow&\mathring{\alpha}_{n}^{(i)}=0;\vspace{2mm}\\\
a=-1,&\epsilon=\delta,&q=0&\Rightarrow&\mathring{\beta}_{n}^{(1)}=\mathring{\beta}_{n}^{(2)}=\mathring{\beta}_{n}^{(7)}=\mathring{\beta}_{n}^{(8)}=0;\vspace{2mm}\\\
a=-1,&\alpha+\beta=1+\gamma,&q=\gamma(1-\delta)&\Rightarrow&\mathring{\beta}_{n}^{(3)}=\mathring{\beta}_{n}^{(4)}=\mathring{\beta}_{n}^{(5)}=\mathring{\beta}_{n}^{(6)}=0.\end{array}$
(117)

For these sets of parameters, the recurrence relations (72) become two-terms
relations and the Heun equation (2) can be reduced to the hypergeometric
equation (4) – see Appendix B.

Finally, by considering the solutions $\mathring{H}^{(i+4)}(x)$ and
$\mathring{H}^{(i)}(x)$ ($i=1,2,3,4$) we find that

$\displaystyle\mbox{for infinite
series,}\quad\mathring{H}^{(i+4)}(x)=\mathring{H}^{(i)}(x)\quad\text{if}\quad|x|<|a|\quad\mbox{and}\quad
1-\epsilon\neq 0,1,2,\cdots.$ (118)

Relations (118) results from [32]

$\displaystyle(1+y)^{\kappa}=\sum_{m=0}^{\infty}\frac{\Gamma(\kappa+m)}{m!\;\Gamma(\kappa)}\;y^{m},\qquad|y|<1,\qquad\kappa\neq
0,1,2,\cdots$ (119)

applied to the factor $[1-x/a]^{1-\epsilon}$, and from the Cauchy product for
infinite series, namely,

$\displaystyle\sum_{n=0}^{\infty}f_{n}\sum_{m=0}^{\infty}g_{m}=\sum_{j=0}^{\infty}h_{j},\quad\mathrm{where}\qquad
h_{j}=\sum_{l=0}^{j}f_{l}\,g_{j-l}.$ (120)

For instance, for $\mathring{H}^{(5)}(x)$ we have

$\displaystyle\begin{array}[]{l}\mathring{H}^{(5)}(x)\stackrel{{\scriptstyle\text{(\ref{S5})}}}{{=}}\displaystyle\sum_{n=0}^{\infty}\mathring{b}_{n}^{(5)}x^{n}\left[1-\frac{x}{a}\right]^{1-\epsilon}\stackrel{{\scriptstyle\text{(\ref{binomial})}}}{{=}}\displaystyle\sum_{n=0}^{\infty}\mathring{b}_{n}^{(5)}x^{n}\sum_{m=0}^{\infty}\left(-\frac{1}{a}\right)^{m}\frac{\Gamma(1-\epsilon+m)}{\Gamma(1-\epsilon)m!}\;x^{m}\vspace{2mm}\\\
\stackrel{{\scriptstyle\text{(\ref{cauchy})}}}{{=}}\displaystyle\sum_{j=0}^{\infty}x^{j}\left[\frac{1}{\Gamma(1-\epsilon)}\sum_{l=0}^{j}\left(-\frac{1}{a}\right)^{j-l}\frac{\Gamma(1-\epsilon+j-l)}{(j-l)!}\;\mathring{b}_{j}^{(5)}\right],\;|x|<|a|,\;\;1-\epsilon\neq
0,1,\cdots.\end{array}$

Thence, by taking $j=n$ and comparing the above equation with (98), we get

$\displaystyle\begin{array}[]{l}\mathring{H}^{(5)}(x)=\mathring{H}^{(1)}(x)\quad\text{
with
}\quad\mathring{b}_{n}^{(1)}=\frac{1}{\Gamma(1-\epsilon)}\displaystyle\sum_{l=0}^{n}\left(-\frac{1}{a}\right)^{n-l}\frac{\Gamma(1-\epsilon+n-l)}{(n-l)!}\mathring{b}_{l}^{(5)}.\end{array}$

In fact, Eqs. (118) generalize an Ince result [33] for solutions of the Lamé
equation (in which case $\gamma=\delta=\epsilon=1/2$, $x=\mathrm{sn}^{2}{u}$
and $a=1/k^{2}$).

Finally we set the limits of $\mathring{H}^{(1)}$ for the confluent Heun
equation (CHE) which, in the form used by Arscott [21], reads

$\displaystyle x(x-1)\frac{d^{2}S(x)}{dx^{2}}+[-\gamma+(\gamma+\delta)x+\rho
x(x-1)]\frac{dS(x)}{dx}+(\alpha\rho x-\sigma)S(x)=0,\vspace{2mm}$ (123)
$\displaystyle\rho\neq 0,\quad(CHE),$

where $\rho$ and $\sigma$ are constants. The points $x=0$ and $x=1$ are
regular singular points while $z=\infty$ is an irregular point. Eq. (123)
results from the Heun equation (2) by the limits

$\displaystyle a,\;\beta,\;q\to\infty\;\;\mbox{such
that}\quad\frac{\beta}{a}\to\frac{\epsilon}{a}\to-\rho,\quad\frac{q}{a}\to-\sigma,\qquad
H(x)\to S(x).$ (124)

Dividing the recurrence relations by $a$ and using the previous limits, the
expansion (98) gives ($\mathring{b}_{-1}^{(1)}=0$)

$\displaystyle\begin{cases}\mathring{S}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\mathring{b}_{n}^{(1)}x^{n},\\\
(n+\gamma)(n+1)\mathring{b}_{n+1}^{(1)}-\left[n^{2}+(\gamma+\delta-1-\rho)n-\sigma\right]\mathring{b}_{n}^{(1)}-\rho(n+\alpha-1)\mathring{b}_{n-1}^{(1)}=0,\end{cases}$
(125)

as solution to the CHE (123). From (125) we can generate other expansions by
means of transformations of the CHE [22, 34]. Actually, expansion (125)
includes the one used in 1935 by Baber and Hassé to obtain convergent
solutions to the angular part of the two-center problem of quantum mechanics
[35].

###  3.2 Series $\mathring{\bm{H}}^{(i)}(x)$ around $x=1$

The following power series $\mathring{\bm{H}}^{(i)}(x)$ around $x=1$ come from
$\mathring{\bm{H}}^{(1)}(x)=M_{49}\mathring{H}^{(i)}(x)$,
$\mathring{\bm{H}}^{(i)}(x)=T_{i}\mathring{\bm{H}}^{(1)}(x)$. In the following
the coefficients $\mathring{\gamma}_{n}^{(i)}$ are the ones of section III.A.

$\displaystyle\begin{cases}\mathring{\bm{H}}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\mathring{\bm{b}}_{n}^{(1)}(1-x)^{n},\\\
\mathring{\bm{\alpha}}_{n}^{(1)}=(1-a)(n+\delta)(n+1),\quad\mathring{\bm{\beta}}_{n}^{(1)}=(a-2)n^{2}+[(a-1)(\gamma+\delta-1)-\alpha-\beta\vspace{2mm}\\\
+\gamma]n+q-\alpha\beta,\qquad\mathring{\bm{\gamma}}_{n}^{(1)}=\mathring{\gamma}_{n}^{(1)}.\end{cases}$
(126)
$\displaystyle\begin{cases}\mathring{\bm{H}}^{(2)}(x)=x^{1-\gamma}\displaystyle\sum_{n=0}^{\infty}\mathring{\bm{b}}_{n}^{(2)}(1-x)^{n},\\\
\mathring{\bm{\alpha}}_{n}^{(2)}=(1-a)(n+\delta)(n+1),\qquad\mathring{\bm{\beta}}_{n}^{(2)}=(a-2)n^{2}+[(a-1)(1-\gamma+\delta)-\alpha\\\
-\beta+\gamma]n+q-a(\gamma-1)\delta+(\gamma-1)\delta-\alpha\beta,\qquad\mathring{\bm{\gamma}}_{n}^{(2)}=\mathring{\gamma}_{n}^{(2)},\end{cases}$
(127)
$\displaystyle\begin{cases}\mathring{\bm{H}}^{(3)}(x)=(1-x)^{1-\delta}\displaystyle\sum_{n=0}^{\infty}\mathring{\bm{b}}_{n}^{(3)}(1-x)^{n},\\\
\mathring{\bm{\alpha}}_{n}^{(3)}=(1-a)(n+2-\delta)(n+1),\quad\mathring{\bm{\beta}}_{n}^{(3)}=(a-2)n^{2}+\big{[}(a-1)(1+\gamma-\delta)\\\
-\alpha-\beta+\gamma+2\delta-2\big{]}n+q-a\gamma(\delta-1)-\alpha\beta+(\delta-1)(\alpha+\beta-\delta+1),\\\
\mathring{\bm{\gamma}}_{n}^{(3)}=\mathring{\gamma}_{n}^{(3)}.\end{cases}$
(128)
$\displaystyle\begin{cases}\mathring{\bm{H}}^{(4)}(x)=x^{1-\gamma}(1-x)^{1-\delta}\displaystyle\sum_{n=0}^{\infty}\mathring{\bm{b}}_{n}^{(4)}(1-x)^{n},\\\
\mathring{\bm{\alpha}}_{n}^{(4)}=(1-a)(n+2-\delta)(n+1),\qquad\mathring{\bm{\beta}}_{n}^{(4)}=(a-2)n^{2}+\big{[}(a-1)(3-\gamma-\delta)\\\
-\alpha-\beta+\gamma+2\delta-2\big{]}n+q-a(\gamma+\delta-2)-\alpha\beta+\gamma+\delta-2+(\delta-1)\epsilon,\\\
\mathring{\bm{\gamma}}_{n}^{(4)}=\mathring{\gamma}_{n}^{(4)}.\end{cases}$
(129)
$\displaystyle\begin{cases}\mathring{\bm{H}}^{(5)}(x)=\displaystyle\left[1-\frac{x}{a}\right]^{1-\epsilon}\sum_{n=0}^{\infty}\mathring{\bm{b}}_{n}^{(5)}(1-x)^{n},\\\
\mathring{\bm{\alpha}}_{n}^{(5)}=(1-a)(n+\delta)(n+1),\qquad\mathring{\bm{\beta}}_{n}^{(5)}=(a-2)n^{2}+\big{[}(a-1)(\gamma+\delta-1)\\\
+\alpha+\beta-\gamma-2\delta\big{]}n+q-\alpha\beta+(\alpha+\beta-\gamma-\delta)\delta,\qquad\mathring{\bm{\gamma}}_{n}^{(5)}=\mathring{\gamma}_{n}^{(5)}.\end{cases}$
(130) $\displaystyle\begin{cases}\mathring{\bm{H}}^{(6)}(x)=\displaystyle
x^{1-\gamma}\left[1-\frac{x}{a}\right]^{1-\epsilon}\sum_{n=0}^{\infty}\mathring{\bm{b}}_{n}^{(6)}(1-x)^{n},\\\
\mathring{\bm{\alpha}}_{n}^{(6)}=(1-a)(n+\gamma)(n+1),\qquad\mathring{\bm{\beta}}_{n}^{(6)}=(a-2)n^{2}+\big{[}(a-1)(1-\gamma+\delta)\\\
+\alpha+\beta-\gamma-2\delta\big{]}n+q-a(\gamma-1)\delta-\alpha\beta+(\alpha+\beta-\delta-1)\delta,\qquad\bm{\gamma}_{n}^{(6)}=\mathring{\gamma}_{n}^{(6)}.\end{cases}$
(131)
$\displaystyle\begin{cases}\mathring{\bm{H}}^{(7)}(x)=\displaystyle(1-x)^{1-\delta}\left[1-\frac{x}{a}\right]^{1-\epsilon}\sum_{n=0}^{\infty}\mathring{\bm{b}}_{n}^{(7)}(1-x)^{n},\\\
\mathring{\bm{\alpha}}_{n}^{(7)}=(1-a)(n+2-\delta)(n+1),\qquad\mathring{\bm{\beta}}_{n}^{(7)}=(a-2)n^{2}+\big{[}(a-1)(1+\gamma-\delta)\\\
+\alpha+\beta-\gamma-2\big{]}n+q-a\gamma(\delta-1)-\alpha\beta+\gamma(\delta-1)+\alpha+\beta-\gamma-1,\\\
\mathring{\bm{\gamma}}_{n}^{(7)}=\mathring{\gamma}_{n}^{(7)}.\end{cases}$
(132) $\displaystyle\begin{cases}\mathring{\bm{H}}^{(8)}(x)=\displaystyle
x^{1-\gamma}(1-x)^{1-\delta}\left[1-\frac{x}{a}\right]^{1-\epsilon}\sum_{n=0}^{\infty}\mathring{\bm{b}}_{n}^{(8)}(1-x)^{n},\\\
\mathring{\bm{\alpha}}_{n}^{(8)}=(1-a)(n+2-\delta)(n+1),\qquad\mathring{\bm{\beta}}_{n}^{(7)}=(a-2)n^{2}+\big{[}(a-1)(3-\gamma-\delta)\\\
+\alpha+\beta-\gamma-2\big{]}n+q+a(2-\gamma-\delta)-\alpha\beta+\alpha+\beta+\delta-3,\qquad\bm{\gamma}_{n}^{(8)}=\mathring{\gamma}_{n}^{(8)}.\end{cases}$
(133)

We find that

$\displaystyle\begin{array}[]{l}a=1\quad\Rightarrow\quad\mathring{\bm{\alpha}}_{n}^{(i)}=0;\vspace{2mm}\\\
a=2,\;\alpha+\beta+1=2\gamma+\delta,\;q=\alpha\beta\quad\Rightarrow\quad\mathring{\bm{\beta}}_{n}^{(1)}=\mathring{\bm{\beta}}_{n}^{(3)}=\mathring{\bm{\beta}}_{n}^{(6)}=\mathring{\bm{\beta}}_{n}^{(8)}=0;\vspace{3mm}\\\
a=2,\;\alpha+\beta=1+\delta,\;q=\alpha\beta+(\gamma-1)\delta\quad\Rightarrow\quad\mathring{\bm{\beta}}_{n}^{(2)}=\mathring{\bm{\beta}}_{n}^{(4)}=\mathring{\bm{\beta}}_{n}^{(5)}=\mathring{\bm{\beta}}_{n}^{(7)}=0\end{array}$
(137)

in the expansions (126-133). Once more, for these cases, the Heun equation (2)
can be reduced to the hypergeometric equation (4) – see Appendix B.

##  4 Solutions in series of hypergeometric functions

This section presents the two groups of solutions in series of hypergeometric
functions, $F(\mathrm{a,b;c};z)$, which are denoted by $H^{(i)}(x)$ and
$\bm{H}^{(i)}(x)$ ($i=1,2,\cdots,8$) and correspond, respectively, to the
series coefficients denoted by ${b}_{n}^{(i)}$ and $\bm{b}_{n}^{(i)}$. The
solutions are generated from the solution $\bar{H}^{(1)}(x)$, written in Eqs.
(141b) and (141d), by means of the following transformations

$\displaystyle{H}^{(1)}(x)=M_{17}\bar{H}^{(1)}(x),\qquad{H}^{(i)}(x)=T_{i}{H}^{(1)}(x),\vspace{2mm}$
(138)
$\displaystyle\bm{H}^{(1)}(x)=M_{49}\bar{H}^{(1)}(x),\qquad\bm{H}^{(i)}(x)=T_{i}\bm{H}^{(1)}(x),$
(139)

where $M_{17}$ and $M_{49}$ are the fractional transformations given in Eqs.
(62) and (64), and $T_{i}$ are the homotopic transformations (A.2-A.20).

In section 4.1 we construct the initial solution $\bar{H}^{1}(x)$). In
sections 4.2 and 4.3 we write the solutions $H^{(i)}(x)$ and
$\bm{H}^{(i)}(x)$. Formally the solutions are given by infinite series, but
they are applied only for the cases in which the series become finite. By this
reason, there is no discussion on convergence of the series.

To simplify the expressions of the recurrence relations, from Eq. (159)
onwards, the solutions are written in terms of the functions
$\tilde{F}(\mathrm{a,b;c};z)$ defined as

$\displaystyle\tilde{F}(\mathrm{a,b;c};z)=\frac{1}{\Gamma(\mathrm{c})}{F}(\mathrm{a,b;c};z)$
(140)

where $\Gamma$ stands for gamma function.

###  4.1 The initial solution $\bar{H}^{(1)}(x)$

The solution $\bar{H}^{(1)}$ in series of hypegeometric functions
$F\left(n+\alpha,\gamma+\delta-\alpha-1;n+\gamma;x\right)$ is

$\displaystyle\begin{array}[]{l}\bar{H}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\bar{b}_{n}^{(1)}x^{n}F\left(n+\alpha,\gamma+\delta-\alpha-1;n+\gamma;x\right).\end{array}$
(141b)

where the recurrence relations (72) for $\bar{b}_{n}^{(1)}$ take the form
$\bar{\alpha}_{n}^{(1)}\bar{b}_{n+1}^{(1)}+\bar{\beta}_{n}^{(1)}\bar{b}_{n}^{(1)}+\bar{\gamma}_{n}^{(1)}\bar{b}_{n-1}^{(1)}=0$
with $\bar{b}_{-1}^{(1)}=0$ and

$\displaystyle\begin{array}[]{l}\bar{\alpha}_{n}^{(1)}=a(n+1)(n+\gamma),\quad\bar{\beta}_{n}^{(1)}=-(a+1)n^{2}-[a(2\alpha+1-\gamma-\delta)+\alpha+\beta-\delta]n\end{array}\vspace{2mm}$
(141d)
$\displaystyle\begin{array}[]{l}-q-a\alpha(\alpha+1-\gamma-\delta),\qquad\bar{\gamma}_{n}^{(1)}=\frac{(n+\alpha-1)(n+\alpha-\delta)(n+\alpha+\beta-\gamma-\delta)}{n+\gamma-1}.\end{array}$
(141f)

The above hypergeometric functions converge for any $|x|<1$ and on $|x|=1$ if
$\mbox{Re }\delta<1$. For this reason, the solution $\bar{H}^{(1)}(x)$ will be
valid only for $|x|\leq 1$ in opposition to $\mathring{H}^{(1)}(x)$ which may
be valid for $|x|>1$ if $|a|>1$ – see conditions (108). In fact,
$\bar{H}^{(1)}(x)=\mathring{H}^{(1)}(x)$ only if $\gamma+\delta-1=\alpha$.

The Heun equation (2) is equivalent to

$(x-a)\left[x(x-1)\frac{d^{2}}{dx^{2}}+(\gamma+\delta)x\frac{d}{dx}-\gamma\frac{d}{dx}\right]H+\epsilon
x(x-1)\frac{dH}{dx}+\left[\alpha\beta x-q\right]H=0,$ (142)

The solution (141b) is obtained by writing

$\displaystyle\begin{array}[]{l}H(x)=\bar{H}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\bar{b}_{n}^{(1)}x^{n}F_{n}^{(1)}(x).\quad\mbox{where}\vspace{2mm}\\\
F_{n}^{(1)}(x)=F\left(\rho_{n},\mathrm{b};\sigma_{n};x\right),\qquad\rho_{n}=n+\alpha,\quad\mathrm{b}=\gamma+\delta-\alpha-1,\quad\sigma_{n}=n+\gamma.\end{array}$
(145)

Then,

$\displaystyle\frac{dH}{dx}=\displaystyle\sum_{n=0}^{\infty}\bar{b}_{n}^{(1)}x^{n}\left[\frac{d}{dx}+\frac{n}{x}\right]F_{n}^{(1)},\quad\frac{d^{2}H}{dx^{2}}=\displaystyle\sum_{n=0}^{\infty}\bar{b}_{n}^{(1)}x^{n}\left[\frac{d^{2}}{dx^{2}}+\frac{2n}{x}\frac{d}{dx}+\frac{n(n-1)}{x^{2}}\right]F_{n}^{(1)}.$

By inserting the previous expressions into Eq. (142) and using the
hypergeometric equation

$\displaystyle
x(1-x)\frac{d^{2}F_{n}^{(1)}(x)}{dx^{2}}+[\sigma_{n}-(\rho_{n}+\mathrm{b}+1)x]\frac{dF_{n}^{(1)}(x)}{dx}-\rho_{n}\mathrm{b}F_{n}^{(1)}(x)=0,$

for $F_{n}(x)$, we find

$\displaystyle\begin{array}[]{l}\displaystyle\sum_{n=0}^{\infty}\bar{b}_{n}^{(1)}x^{n}\left[an(1-x)\frac{dF_{n}^{(1)}}{dx}+(n+\epsilon)x(x-1)\frac{dF_{n}^{(1)}}{dx}\right]\end{array}+$
(147)
$\displaystyle\begin{array}[]{l}\displaystyle\sum_{n=0}^{\infty}\bar{b}_{n}^{(1)}x^{n+1}\left[n(n+\gamma+\delta-1+\epsilon)+\alpha\beta-\mathrm{b}\rho_{n}\right]F_{n}^{(1)}+\end{array}$
(149)
$\displaystyle\begin{array}[]{l}\displaystyle\sum_{n=0}^{\infty}\bar{b}_{n}^{(1)}x^{n}\left[a\alpha(\gamma+\delta-\alpha-1)-q-an(n+\alpha)-n(n+\gamma-1+\epsilon)\right]F_{n}^{(1)}+\end{array}$
(151)
$\displaystyle\begin{array}[]{l}\displaystyle\sum_{n=0}^{\infty}\bar{b}_{n}^{(1)}an(n-1+\gamma)x^{n-1}F_{n}^{(1)}=0.\end{array}$
(153)

On the other side, from the properties (57) of the hypergeometric functions we
get

$\displaystyle\begin{array}[]{l}(1-x)\frac{dF_{n}^{(1)}}{dx}=\left(\mathrm{b}-\frac{\sigma_{n}-1}{x}\right)F_{n}^{(1)}+\frac{\sigma_{n-1}}{x}F_{n-1}^{(1)},\qquad
x(x-1)\frac{dF_{n}^{(1)}}{dx}=\end{array}\vspace{2mm}$ (155)
$\displaystyle\begin{array}[]{l}-\rho_{n}xF_{n}^{(1)}+\frac{\rho_{n}(\sigma_{n}-\mathrm{b})}{\sigma_{n}}xF_{n+1}^{(1)}\end{array}.$
(157)

By means of these equations, Eq. (147) becomes

$\displaystyle\displaystyle\sum_{n=1}^{\infty}\bar{b}_{n}^{(1)}\sigma_{n-1}x^{n-1}\
F_{n-1}^{(1)}+\displaystyle\sum_{n=0}^{\infty}\bar{b}_{n}^{(1)}x^{n}\bar{\beta}_{n}F_{n}^{(1)}+\displaystyle\sum_{n=0}^{\infty}\bar{b}_{n}^{(1)}\frac{\rho_{n}(\sigma_{n}-\mathrm{b})(n+\epsilon)}{\sigma_{n}}x^{n+1}F_{n+1}^{(1)}=0.$

Now, by setting $n=m+1$ and $n=m-1$ in the first and last terms of the above
equation, one finds

$\displaystyle\left[a\sigma_{0}\;\bar{b}_{1}^{(1)}+\bar{\beta}_{0}\;\bar{b}_{0}^{(1)}\right]F_{0}^{(1)}(x)+$
$\displaystyle\displaystyle\sum_{n=1}^{\infty}\begin{array}[]{l}\left[a(n+1)\sigma_{n}\;\bar{b}_{n+1}^{(1)}+\bar{\beta}_{n}\;\bar{b}_{n}^{(1)}+\frac{\rho_{n-1}(\sigma_{n-1}-\mathrm{b})(n+\epsilon-1)}{\sigma_{n-1}}\;\bar{b}_{n-1}^{(1)}\right]x^{n}F_{n}^{(1)}(x)=0.\end{array}$

Thence, the recurrence relations for $\bar{b}_{n}^{(1)}$ follow by requiring
that the coefficients of the functions $x^{n}F_{n}^{(1)}(x)$ vanish.

Finally, by taking $\tilde{b}_{n}^{(1)}=\Gamma(n+\gamma)\bar{b}_{n}$, the
expansion $\bar{H}^{(1)}$ is re-expressed in terms of the functions
$\tilde{F}$ defined in (140), namely,

$\displaystyle\begin{cases}\bar{H}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\tilde{b}_{n}^{(1)}x^{n}\tilde{F}\left(n+\alpha,\gamma+\delta-\alpha-1;n+\gamma;x\right)\quad\mbox{where,
in (\ref{recurrence1} ) for }\tilde{b}_{n}^{(1)},\vspace{3mm}\\\
\textasciidieresis\tilde{\alpha}_{n}^{(1)}=a(n+1),\qquad\tilde{\beta}_{n}^{(1)}=-(a+1)n^{2}-[a(2\alpha+1-\gamma-\delta)+\alpha+\beta-\delta]n-q\vspace{2mm}\\\
-a\alpha(\alpha+1-\gamma-\delta),\qquad\tilde{\gamma}_{n}^{(1)}=(n+\alpha-1)(n+\alpha-\delta)(n+\alpha+\beta-\gamma-\delta).\end{cases}$
(159)

Other solutions are given $\bar{H}^{(i)}(x)=T_{i}\bar{H}^{(1)}(x)$. Some of
these expansions yield finite-series solutions for the problem discussed in
Section 5; however, such solutions do not fulfil the Arscott conditions (82).
By this reason, we use $\bar{H}^{(1)}(x)$ to obtain the expansions
${H}^{(i)}(x)$ and ${\bm{H}}^{(i)}(x)$ which are used in Sections 5.3 and 5.4,
respectively.

As a by-product we find that the expansion (159) admits the limits (124) for
the CHE (123), leading to

$\displaystyle\begin{cases}\bar{S}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\tilde{b}_{n}^{(1)}x^{n}\tilde{F}\left(n+\alpha,\gamma+\delta-\alpha-1;n+\gamma;x\right)\qquad\mbox{with}\vspace{3mm}\\\
(n+1)\;\tilde{b}_{n+1}^{(1)}-\left[n^{2}+(2\alpha+1-\gamma-\delta-\rho)n-\sigma+\alpha(\alpha+1-\gamma-\delta)\right]\;\tilde{b}_{n}^{(1)}-\vspace{3mm}\\\
\rho(n+\alpha-1)(n+\alpha-\delta)\;\tilde{b}_{n-1}^{(1)}=0,\qquad\left[\tilde{b}_{-1}^{(1)}=0\right].\end{cases}$
(160)

This $\bar{S}^{(1)}$ is equivalent to a solution for the CHE proposed 1937 by
E. Fisher [36].

###  4.2 The solutions ${H}^{(i)}(x)$ around ${(a-1)x}/{(a-x)}=0$

In this section ${{H}}^{(1)}(x)=M_{17}\bar{H}^{(1)}(x)$ and
${{H}}^{(i)}(x)=T_{i}{{H}}^{(1)}(x)$, where $\bar{H}^{(1)}$ is given in (159).
The recurrence relations for the coefficients ${b}_{n}^{(i)}$ once more have
the form (72).

$\displaystyle\begin{cases}{{H}}^{(1)}(x)=\left(1-\frac{x}{a}\right)^{-\alpha}\displaystyle\sum_{n=0}^{\infty}\begin{array}[]{l}b_{n}^{(1)}\left[\frac{(a-1)x}{a-x}\right]^{n}\tilde{F}\left[n+\alpha,\gamma+\delta-\alpha-1;n+\gamma;\frac{(a-1)x}{a-x}\right],\vspace{2mm}\end{array}\\\
\textasciidieresis{{\alpha}}_{n}^{(1)}=(1-a)(n+1),\qquad{\beta}_{n}^{(1)}=(a-2)n^{2}+[(a-1)(2\alpha+1-\gamma-\delta)-\alpha+\vspace{3mm}\\\
\beta-\gamma]n+q+(a-1)\alpha(\alpha+1-\gamma-\delta)-\alpha\gamma,\qquad{\gamma}_{n}^{(1)}=(n+\alpha-1)\times\\\
\hskip 22.76219pt(n+\alpha-\delta)(n+\alpha-\beta).\end{cases}$ (161)
$\displaystyle\begin{cases}{H}^{(2)}(x)=x^{1-\gamma}\left(1-\frac{x}{a}\right)^{\gamma-\beta-1}\times\vspace{2mm}\\\
\hskip
51.21504pt\displaystyle\sum_{n=0}^{\infty}\begin{array}[]{l}{b}_{n}^{(2)}\left[\frac{(a-1)x}{a-x}\right]^{n}\tilde{F}\left[n+\beta+1-\gamma,\delta-\beta;n+2-\gamma;\frac{(a-1)x}{a-x}\right]\end{array},\vspace{3mm}\\\
{\alpha}_{n}^{(2)}=(1-a)(n+1),\qquad{\beta}_{n}^{(2)}=(a-2)n^{2}+[(a-1)(2\beta+1-\gamma-\delta)-2+\vspace{3mm}\\\
\alpha-\beta+\gamma]n+q+a\beta(\beta+1-\gamma-\delta)-\beta(\beta-\delta+1)+(\gamma-1)(1-\alpha+\beta),\vspace{3mm}\\\
{\gamma}_{n}^{(2)}={(n+\beta-\gamma)(n+\beta+1-\gamma-\delta)(n-\alpha+\beta)}\end{cases}$
(162)

The solutions ${H}^{(3)}$ and ${H}^{(4)}$ take the forms

$\displaystyle{H}^{(3)}(x)=(1-x)^{1-\delta}\left(1-\frac{x}{a}\right)^{\delta-\beta-1}\times\vspace{2mm}$
$\displaystyle\displaystyle\sum_{n=0}^{\infty}\begin{array}[]{l}{b}_{n}^{(3)}\left[\frac{(a-1)x}{a-x}\right]^{n}\tilde{F}\left[n+\beta+1-\delta,\gamma-\beta;n+\gamma;\frac{(a-1)x}{a-x}\right]\end{array}\stackrel{{\scriptstyle\text{(\ref{euler-1}})}}{{=}}{H}^{(1)}(x)\Big{|}_{\alpha\leftrightarrow\beta},\qquad$
(164)
$\displaystyle{H}^{(4)}(x)=x^{1-\gamma}(1-x)^{1-\delta}\left(1-\frac{x}{a}\right)^{\gamma+\delta-\alpha-2}\displaystyle\sum_{n=0}^{\infty}{b}_{n}^{(4)}\times\vspace{2mm}$
$\displaystyle\begin{array}[]{l}\left[\frac{(a-2)x}{a-x}\right]^{n}\tilde{F}\left[n+2+\alpha-\gamma-\delta,1-\alpha;n+2-\gamma;\frac{(a-1)x}{a-x}\right]\end{array}{=}{H}^{(2)}(x)\Big{|}_{\alpha\leftrightarrow\beta},$
(166)

where $\alpha$ and $\beta$ must be interchanged also in the recurrence
relations.

$\displaystyle\begin{cases}{H}^{(5)}(x)=\left(1-\frac{x}{a}\right)^{-\beta}\displaystyle\sum_{n=0}^{\infty}\begin{array}[]{l}{b}_{n}^{(5)}\left[\frac{(a-1)x}{a-x}\right]^{n}\tilde{F}\left[n-\alpha+\gamma+\delta,\alpha-1;n+\gamma;\frac{(a-1)x}{a-x}\right],\end{array}\\\
{\alpha}_{n}^{(5)}=(1-a)(n+1),\qquad{\beta}_{n}^{(5)}=(a-2)n^{2}+[(a-1)(1-2\alpha+\gamma+\delta)+\vspace{3mm}\\\
\alpha-\beta-\gamma]n+q+(a-1)(\alpha-1)(\alpha-\gamma-\delta)-\beta\gamma,\vspace{3mm}\\\
{\gamma}_{n}^{(5)}={(n-\alpha+\gamma+\delta-1)(n-\alpha+\gamma)(n-\alpha+\beta)}.\end{cases}$
(167)
$\displaystyle\begin{cases}{H}^{(6)}(x)=x^{1-\gamma}\left(1-\frac{x}{a}\right)^{\gamma-\alpha-1}\displaystyle\sum_{n=0}^{\infty}\begin{array}[]{l}{b}_{n}^{(6)}\left[\frac{(a-1)x}{a-x}\right]^{n}\end{array}\times\vspace{2mm}\\\
\hskip
142.26378pt\begin{array}[]{l}\tilde{F}\left[n-\beta+\delta+1,\beta-\gamma;n+2-\gamma;\frac{(a-1)x}{a-x}\right],\end{array}\vspace{3mm}\\\
{\alpha}_{n}^{(6)}=(1-a)(n+1),\quad{\beta}_{n}^{(6)}=(a-2)n^{2}+[(a-1)(1-2\beta+\gamma+\delta)-\alpha+\vspace{3mm}\\\
\beta+\gamma-2]n+q-\alpha+a(\beta-1)(\beta-\gamma-\delta)-(\beta-1)(\beta-\delta-1),\qquad\vspace{3mm}\\\
{\gamma}_{n}^{(6)}={(n-\beta+\delta)(n-\beta+1)(n+\alpha-\beta)}.\end{cases}$
(168)

The solutions ${H}^{(7)}(x)$ and ${H}^{(8)}(x)$ are expressed as

$\displaystyle{H}^{(7)}(x)=\begin{array}[]{l}(1-x)^{1-\delta}\left(1-\frac{x}{a}\right)^{\delta-\alpha-1}\end{array}\times$
(170)
$\displaystyle\displaystyle\sum_{n=0}^{\infty}\begin{array}[]{l}{b}_{n}^{(7)}\left[\frac{(a-1)x}{a-x}\right]^{n}\tilde{F}\left[n+1-\beta+\gamma,\beta-\delta;n+\gamma;\frac{(a-1)x}{a-x}\right]\stackrel{{\scriptstyle\text{(\ref{euler-1}})}}{{=}}H^{(5)}(x)\Big{|}_{\alpha\leftrightarrow\beta},\end{array}$
(172)
$\displaystyle{H}^{(8)}(x)=\begin{array}[]{l}x^{1-\gamma}(1-x)^{1-\delta}\left(1-\frac{x}{a}\right)^{\gamma+\delta-\beta-2}\displaystyle\sum_{n=0}^{\infty}{b}_{n}^{(8)}\times\end{array}$
(174)
$\displaystyle\begin{array}[]{l}\left[\frac{(a-1)x}{a-x}\right]^{n}\tilde{F}\left[n+2-\alpha,\alpha+1-\gamma-\delta;n+2-\gamma;\frac{(a-1)x}{a-x}\right]\end{array}\stackrel{{\scriptstyle\text{(\ref{euler-1}})}}{{=}}H^{(6)}(x)\Big{|}_{\alpha\leftrightarrow\beta}.$
(176)

In this group the hypergeometric functions converge in the region
$|{(a-1)x}/{(a-x)}|<1$ which includes only one singular point ($x=0$); in
accordance with (47), a restriction on $\delta$ assure convergence also on
$|{(a-1)x}/{(a-x)}|=1$ (includes $x=1$). Precisely,

$\displaystyle\begin{array}[]{l}\mbox{hypergeometric functions in
}{H}^{(i)}\text{ converge on }\left|\frac{(a-1)x}{(a-x)}\right|=1\mbox{ if
}\mbox{Re}\;\delta<1.\end{array}$ (178)

In fact, in view of (49) there are two forms for each expansion in series of
hypergeometric functions, but only one of them is suitable for $|z|=1$. For
example,

$\displaystyle{{H}}^{(1)}(x)=\left(1-\frac{x}{a}\right)^{-\alpha}\displaystyle\sum_{n=0}^{\infty}\begin{array}[]{l}b_{n}^{(1)}\left[\frac{(a-1)x}{a-x}\right]^{n}\tilde{F}\left[n+\alpha,\gamma+\delta-\alpha-1;n+\gamma;\frac{(a-1)x}{a-x}\right]\end{array}\vspace{2mm}$
$\displaystyle\stackrel{{\scriptstyle\text{(\ref{euler-1}})}}{{=}}(1-x)^{1-\delta}\left(1-\frac{x}{a}\right)^{\delta-1-\alpha}\displaystyle\sum_{n=0}^{\infty}\begin{array}[]{l}b_{n}^{(1)}\left[\frac{(a-1)x}{a-x}\right]^{n}\tilde{F}\left[n+1+\alpha-\delta,\gamma-\alpha;n+\gamma;\frac{(a-1)x}{a-x}\right].\end{array}$

In the first expression, the hypergeometric function converges on $x=1$ if
$\mbox{Re}\;\delta<1$ while the factor $(1-x/a)^{-\alpha}$ is finite because
$a\neq 0,1$. In the second expression, the hypergeometric function converges
on $x=1$ only if $\mbox{Re}\;\delta>1$, but in this case the factor
$(1-x)^{1-\delta}$ in general becomes infinite at $x=1$. Thus, the second form
is inappropriate to get the behaviour at $x=1$.

We find that, for the expansions ${H}^{(i)}(x)$,

$\displaystyle\begin{array}[]{l}a=1\;\quad\Rightarrow\quad{\alpha}_{n}^{(i)}=0;\vspace{2mm}\\\
a=2,\;\alpha+\beta+1=2\gamma+\delta,\;q=\alpha\beta\quad\Rightarrow\quad{\beta}_{n}^{(1)}={\beta}_{n}^{(3)}={\beta}_{n}^{(6)}={\beta}_{n}^{(8)}=0;\vspace{3mm}\\\
a=2,\;\alpha+\beta=1+\delta,\;q=\alpha\beta+(\gamma-1)\delta\quad\Rightarrow\quad{\beta}_{n}^{(2)}={\beta}_{n}^{(4)}={\beta}_{n}^{(5)}={\beta}_{n}^{(7)}=0,\end{array}$
(184)

which reproduce the conditions given in (137) for reducing the Heun equation
to a hypergeometric equation.

###  4.3 The solutions ${\bm{H}}^{(i)}(x)$ around $x=1$

These arise from the solution $\bar{H}^{(1)}(x)$, given in (159), by means of
the following transformations: ${\bm{H}}^{(1)}(x)=M_{49}\bar{H}^{(1)}(x)$ and
${\bm{H}}^{(i)}(x)=T_{i}{\bm{H}}^{(1)}(x)$.

$\displaystyle\begin{cases}{\bm{H}}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}{\bm{b}}_{n}^{(1)}(1-x)^{n}\tilde{F}\left(n+\alpha,\gamma+\delta-\alpha-1;n+\delta;1-x\right),\vspace{2mm}\\\
{\bm{\alpha}}_{n}^{(1)}=(1-a)(n+1),\qquad{\bm{\beta}}_{n}^{(1)}=(a-2)n^{2}+[(a-1)(2\alpha+1-\gamma-\delta)-\vspace{3mm}\\\
\qquad\alpha-\beta+\gamma]n+q+a\alpha(\alpha+1-\gamma-\delta)-\alpha\epsilon,\\\
{\bm{\gamma}}_{n}^{(1)}={(n+\alpha-1)(n+\alpha-\gamma)(n+\alpha+\beta-\gamma-\delta)}.\end{cases}$
(185) $\displaystyle{\bm{H}}^{(2)}(x)$ $\displaystyle=$ $\displaystyle
x^{1-\gamma}\displaystyle\sum_{n=0}^{\infty}{\bm{b}}_{n}^{(2)}(1-x)^{n}\tilde{F}\left(n+\beta+1-\gamma,\delta-\beta;n+\delta;1-x\right)$
(186) $\displaystyle\stackrel{{\scriptstyle\text{(\ref{euler-1}})}}{{=}}$
$\displaystyle{\bm{H}}^{(1)}(x)\Big{|}_{\alpha\leftrightarrow\beta}.$
$\displaystyle\begin{cases}{\bm{H}}^{(3)}(x)=(1-x)^{1-\delta}\displaystyle\sum_{n=0}^{\infty}{\bm{b}}_{n}^{(3)}(1-x)^{n}\times\\\
\hskip
156.49014pt\tilde{F}\left(n+1+\beta-\delta,\gamma-\beta;n+2-\delta;1-x\right),\vspace{2mm}\\\
{\bm{\alpha}}_{n}^{(3)}=(1-a)(n+1),\quad{\bm{\beta}}_{n}^{(3)}=(a-2)n^{2}+[(a-1)(1+2\beta-\gamma-\delta)-\alpha-\vspace{2mm}\\\
\beta+\gamma+2\delta-2]n+q+a\beta(\beta-\gamma-\delta+1)-(\beta-\delta+1)(\alpha+\beta+1-\gamma-\delta),\quad\vspace{2mm}\\\
{\bm{\gamma}}_{n}^{(3)}={(n+\beta-\delta)(n+1+\beta-\gamma-\delta)(n+\alpha+\beta-\gamma-\delta)},\end{cases}$
(187)
$\displaystyle{\bm{H}}^{(4)}(x)=x^{1-\gamma}(1-x)^{1-\delta}\displaystyle\sum_{n=0}^{\infty}{\bm{b}}_{n}^{(4)}(1-x)^{n}\times$
$\displaystyle\hskip
45.5244pt\tilde{F}\left(n+2+\alpha-\gamma-\delta,1-\alpha;n+2-\delta;1-x\right)\stackrel{{\scriptstyle\text{(\ref{euler-1}})}}{{=}}{\bm{H}}^{(3)}(x)\Big{|}_{\alpha\leftrightarrow\beta}.$
(188)
$\displaystyle\begin{cases}{\bm{H}}^{(5)}(x)=\left(1-\frac{x}{a}\right)^{1-\epsilon}\displaystyle\sum_{n=0}^{\infty}{\bm{b}}_{n}^{(5)}(1-x)^{n}\tilde{F}\left(n-\alpha+\gamma+\delta,\alpha-1;n+\delta;1-x\right),\vspace{2mm}\\\
\bm{\alpha}_{n}^{(5)}=(1-a)(n+1),\quad{\bm{\beta}}_{n}^{(5)}=(a-2)n^{2}+[(a-1)(1-2\alpha+\gamma+\delta)+\alpha+\vspace{2mm}\\\
\beta-\gamma-2\delta]n+q+a(\alpha-\gamma-\delta)(\alpha-1)-(\alpha-\delta)(\alpha+\beta-\gamma-\delta-1)-\gamma,\quad\vspace{2mm}\\\
{\bm{\gamma}}_{n}^{(5)}={(n-1-\alpha+\gamma+\delta)(n-\alpha+\delta)(n-\alpha-\beta+\gamma+\delta)}.\end{cases}$
(189) $\displaystyle{\bm{H}}^{(6)}(x)$ $\displaystyle=$ $\displaystyle
x^{1-\gamma}\left(1-\frac{x}{a}\right)^{1-\epsilon}\displaystyle\sum_{n=0}^{\infty}{\bm{b}}_{n}^{(6)}(1-x)^{n}\tilde{F}\left(n+1-\beta+\delta,\beta-\gamma;n+\delta;1-x\right)$
(190) $\displaystyle\stackrel{{\scriptstyle\text{(\ref{euler-1}})}}{{=}}$
$\displaystyle{\bm{H}}^{(5)}(x)\Big{|}_{\alpha\leftrightarrow\beta}.$
$\displaystyle\begin{cases}{\bm{H}}^{(7)}(x)=(1-x)^{1-\delta}\left[1-\frac{x}{a}\right]^{1-\epsilon}\displaystyle\sum_{n=0}^{\infty}{\bm{b}}_{n}^{(7)}(1-x)^{n}\times\\\
\hskip
48.36967pt\tilde{F}\left(n-\beta+\gamma+1,\beta-\delta;n+2-\delta;1-x\right),\vspace{2mm}\\\
{\bm{\alpha}}_{n}^{(7)}=(1-a)(n+1),\quad{\bm{\beta}}_{n}^{(7)}=(a-2)n^{2}+[(a-1)(1-2\beta+\gamma+\delta)+\alpha+\vspace{2mm}\\\
\beta-\gamma-2]n+q+a(\beta-\gamma-\delta)(\beta-1)-(\beta-1)(\alpha+\beta-\gamma-\delta-1)-\gamma,\quad\vspace{2mm}\\\
{\bm{\gamma}}_{n}^{(7)}={(n-\beta+\gamma)(n+1-\beta)(n-\alpha-\beta+\gamma+\delta)}.\end{cases}$
(191)
$\displaystyle\begin{array}[]{l}{\bm{H}}^{(8)}(x)=x^{1-\gamma}(1-x)^{1-\delta}\left[1-\frac{x}{a}\right]^{1-\epsilon}\displaystyle\sum_{n=0}^{\infty}{\bm{b}}_{n}^{(8)}(1-x)^{n}\end{array}\times$
(193) $\displaystyle\hskip
42.67912pt\begin{array}[]{l}\tilde{F}\left(n+2-\alpha,\alpha+1-\gamma-\delta;n+2-\delta;1-x\right)\end{array}\stackrel{{\scriptstyle\text{(\ref{euler-1}})}}{{=}}{\bm{H}}^{(7)}(x)\Big{|}_{\alpha\leftrightarrow\beta}.$
(195)

The hypergeometric functions in ${\bm{H}}^{(i)}(x)$ converge in the region
$|x-1|<1$ but, by (47),

$\displaystyle\begin{array}[]{l}\mbox{hypergeometric }\mbox{functions in
}{\bm{H}}^{(i)}\text{ converge on }|x-1|=1\mbox{ if
}\mbox{Re}\;\gamma<1.\end{array}$ (197)

For the above expansions, $\bm{H}^{(i)}(x)$,

$\displaystyle\begin{array}[]{lllll}a=1&\qquad\Rightarrow&\bm{\alpha}_{n}^{(i)}=0;\vspace{2mm}\\\
a=2,&\beta=\alpha+1-\delta,&q=\alpha\gamma&\Rightarrow&\bm{\beta}_{n}^{(1)}=\bm{\beta}_{n}^{(3)}=\bm{\beta}_{n}^{(6)}=\bm{\beta}_{n}^{(8)}=0;\vspace{3mm}\\\
a=2,&\alpha=\beta+1-\delta,&q=\beta\gamma&\Rightarrow&\bm{\beta}_{n}^{(2)}=\bm{\beta}_{n}^{(4)}=\bm{\beta}_{n}^{(5)}=\bm{\beta}_{n}^{(7)}=0.\end{array}$
(201)

Thence, Eqs.(72) become two-term recurrence relations for the series
coefficients and the Heun equation (2) is reducible to the hypergeometric
equation (4) (Appendix B).

##  5 Schrödinger equation for associated Lamé potentials

There are some periodic potentials for which the Schrödinger equation leads to
equations of Heun family [17, 18, 19, 20, 37, 38]. Solutions for these
equations in general demand numerical computations. In this section we
consider only some analytical aspects concerning the solutions of the
Schrödinger equation for the associated Lamé potential (37) which depends on
the parameters $\bm{m}$ and $\bm{l}$.

We will find that the solutions of the general Heun equation allow to solve
this problem as follows:

  * •

if $\bm{m}-\bm{l}$ or $\bm{l}+\bm{m}$ is an integer, we get finite series from
the power series expansions given in Section 3; further restrictions on
$\bm{m}$ and $\bm{l}$ assure that these solutions satisfy the Arscott
criterion (82);

  * •

if either $\bm{l}$ or $\bm{m}$ is an integer and the other is half an odd
integer, we get finite series from the expansions in series of hypergeometric
functions given in Section 4; such solutions are degenerate and satisfy the
Arscott criterion only for special values of $\bm{l}$ and $\bm{m}$;

  * •

bounded infinite-series solutions are obtained from the expansions given in
section 3.1.

The finite-series solutions are valid under general restrictions on the
parameters, that is, without regarding particular values for $\bm{m}$ and
$\bm{l}$ (some specific values are discussed in Appendix C). Initially, we
explain how we obtain the several types of solutions and set up some notations
for the solutions.

The Schrödinger equation (28) for the associated Lamé potencial (37) is

$\displaystyle\frac{d^{2}\psi(u)}{du^{2}}+\left[{\cal
E}-\bm{m}(\bm{m}+1)k^{2}\;\mathrm{sn}^{2}u-\bm{l}(\bm{l}+1)k^{2}\;\frac{\mathrm{cn}^{2}u}{\mathrm{dn}^{2}u}\right]\psi(u)=0,$
(202)

which is the Darboux Eq. (18) with parameters ($a=1/k^{2}$)

$\displaystyle\begin{array}[]{l}\alpha=\frac{\bm{l}-\bm{m}+1}{2},\quad\beta=\frac{\bm{l}+\bm{m}+2}{2},\quad\gamma=\delta=\frac{1}{2},\quad
q=\frac{(\bm{l}+1)^{2}}{4}-\frac{{\cal
E}}{4k^{2}},\quad\left(\epsilon=\bm{l}+\frac{3}{2}\right).\end{array}$ (204)

Thence, the eigenfunctions $\psi(u)$ follow from solutions $H(x)$ of the Heun
equation through

$\displaystyle\psi(u)\stackrel{{\scriptstyle\text{Eq.(\ref{substituicoes-2})}}}{{=\joinrel=}}\mathrm{dn}^{\bm{l}+1}u\;H[x(u)],\qquad
0\leq x(u)={\rm sn}^{2}u\leq 1,$ (205)

provided that $H(x)$ is a convergent series and $\psi(u)$ is bounded for any
value of $u$. In addition, we impose the Arscott condition
$\alpha_{n-1}\gamma_{n}>0$ for finite series with two or more terms ($n\geq
1$).

The associated Lamé equation (202) remains invariant under the simultaneous
substitutions

$\displaystyle\bm{m}\mapsto-\bm{m}-1,\qquad\bm{l}\mapsto-\bm{l}-1\qquad(\mbox{leave
Eq. (\ref{associado-gan}) invariant})$ (206)

which also leave invariant the solutions constructed according to (205), as we
will see. Alternatively, (206) can be used to set up a new solution out of a
given solution. Further, according to (69), the interchange of $\bm{l}$ and
$\bm{m}$ in conjunction with the shift of $u$ to $u+K$ leaves the associated
Lamè equation invariant, that is,

$\displaystyle u\mapsto
u+K,\qquad\bm{m}\leftrightarrow\bm{l}\qquad(\mbox{leave Eq. (\ref{associado-
gan}) invariant}).$ (207)

This substitutions may be used to get new solutions to the associated Lamé
equation by means of the relations [3, 19]

$\displaystyle\begin{array}[]{l}\mathrm{sn}(u+K)=\mathrm{cd}\;u,\quad\mathrm{cn}(u+K)=-\sqrt{1-k^{2}}\;\mathrm{sd}\;u,\quad\mathrm{dn}(u+K)=\frac{\sqrt{1-k^{2}}}{\mathrm{dn}\;u}.\end{array}$
(209)

Thence, power series of $\operatorname{sn}{u}$ and $\operatorname{cn}{u}$ are
transformed, respectively, in power series of $\operatorname{cd}{u}$ and
$\operatorname{sd}{u}$. To obtain these series directly from solutions of the
Heun equation it would be necessary to construct two additional groups by the
fractional transformations $M_{17}$ and $M_{65}$ as we see from (69). Notice
that the particular cases

$\displaystyle\begin{array}[]{l}\frac{d^{2}\psi(u)}{du^{2}}+\left[{\cal
E}-\bm{l}(\bm{l}+1)k^{2}\;\frac{\mathrm{cn}^{2}u}{\mathrm{dn}^{2}u}\right]\psi(u)=0,\qquad[\bm{m}=0\mbox{
or }\bm{m}=-1],\quad\mbox{and}\end{array}\vspace{3mm}$ (211)
$\displaystyle\begin{array}[]{l}\frac{d^{2}\psi(u)}{du^{2}}+\left[{\cal
E}-\bm{m}(\bm{m}+1)k^{2}\;\mathrm{sn}^{2}u\right]\psi(u)=0,\quad[\bm{l}=0\mbox{
or }\bm{l}=-1:\mbox{Lam\'{e} equation}],\end{array}$ (213)

are connected by (207). We will see that finite-series solutions which satisfy
the Arscott condition for the first case do not satisfy for the second case,
and vice-versa.

For comparison purposes, we write the Khare-Sukhatme notation [19, 20] for the
parameters $\bm{l}$, $\bm{m}$ and $k^{2}$, namely,

$\displaystyle\bm{l}=b,\qquad\bm{m}=a,\qquad k^{2}=m:\qquad\mbox{Khare-
Sukhatme \cite[cite]{[\@@bibref{}{khare,khare2}{}{}]}}.$ (214)

According to (75), finite series result when $\gamma_{n}=0$ for some value of
$n$. For each power-series expansion the condition $\gamma_{n}=0$ gives two
possibilities for truncating the series on the right-hand side. For solutions
obtained from $\mathring{{H}}^{(1)}$ and $\mathring{\bm{H}}^{(1)}$, the two
possibilities result from the fact that
($\mathring{{\gamma}}_{n}^{(1)}=\mathring{\bm{\gamma}}_{n}^{(1)}$)

$\displaystyle\begin{array}[]{l}\mathring{{\gamma}}_{n}^{(1)}=\left(n-\frac{\bm{m}-\bm{l}+1}{2}\right)\left(n+\frac{\bm{l}+\bm{m}}{2}\right)=0\mbox{
if }\begin{cases}\bm{m}-\bm{l}=1,3,5,\cdots\;\Rightarrow\;0\leq
n\leq\frac{\bm{m-l}-1}{2}\;\mbox{or}\vspace{2mm}\\\
\bm{m}+\bm{l}=-2,-4,\cdots\;\Rightarrow\;0\leq
n\leq-\frac{\bm{m+l}+2}{2}.\end{cases}\end{array}$ (216)

For $\bm{m}=-1/2$, the two cases become identical. For $\bm{m}\neq-1/2$, if
the above conditions on $\bm{m-l}$ and $\bm{m+l}$ are satisfied for the same
values of $\bm{m}$ and $\bm{l}$, thence we have to consider only the
truncation which gives the small interval for $n$. The conditions for this
will be written in relations (224g), (225h), $\cdots$, (231g). Such conditions
imply that both $\bm{l}$ and $\bm{m}$ are half an integer number and,
consequently, they do not stand for solutions of Eqs. (211) and (213).

Solutions resulting from $\mathring{{H}}^{(i)}$ and $\mathring{\bm{H}}^{(i)}$
are expansions in series of $\mathrm{sn}^{2}u$ and $\mathrm{cn}^{2}u$,
respectively. We will find that finite series in terms of $\mathrm{sn}^{2}u$
and $\mathrm{cn}^{2}u$ cannot satisfy simultaneously the Arscott condition
because this condition requires that $\mathring{{\gamma}}_{n}^{(i)}>0$ and
$\mathring{{\gamma}}_{n}^{(i)}<0$, respectively. For the present case, the
Arscott condition is important for assuring that the characteristic equation
(81) yields real and distinct values for the energy ${\cal E}$.

We use the following notations for the solutions of Eq. (202):

  * (1)

the pairs $\left(\mathring{{\psi}}^{(i)},\tilde{{\psi}}^{(i)}\right)$ denote
finite series of $\mathrm{sn}^{2}u,$ obtained from truncation of the powers
series $\mathring{{H}}^{(i)}$ ;

  * (2)

the pairs $\left(\mathring{\bm{\Psi}}^{(i)},\tilde{\bm{\Psi}}^{(i)}\right)$
denote finite series of $\mathrm{cn}^{2}u,$ obtained from truncation of the
powers series $\mathring{\bm{H}}^{(i)}$ ;

  * (3)

$\psi^{(i)}$ ($i=1,2,\cdots,8$) denote finite series of hypergeometric
functions obtained from expansions $H^{(i)}$ of Sec. 4.2;

  * (4)

$\bm{\Psi}^{(i)}$ denote finite series of hypergeometric functions resulting
from expansions $\bm{H}^{(i)}$ of Sec. 4.3;

  * (5)

$\mathring{\Phi}^{(i)}$ denote infinite-series solutions in terms of
$\mathrm{sn}^{2}u$ whose linear dependence requires further investigation.

Each of the above items are discussed in the next subsections. We will use the
following parity and periodicity properties for elliptic functions [3]:

$\displaystyle\begin{array}[]{l}\mathrm{sn}(-u)=-\mathrm{sn}{\;u},\qquad\qquad\mathrm{cn}(-u)=\mathrm{cn}{\;u},\qquad\qquad\;\mathrm{dn}(-u)=\mathrm{dn}{\;u}.\end{array}\vspace{2mm}$
(218)
$\displaystyle\begin{array}[]{l}\mathrm{sn}(u+2K)=-\mathrm{sn}{\;u},\qquad\mathrm{cn}(u+2K)=-\mathrm{cn}{\;u},\qquad\mathrm{dn}(u+2K)=\mathrm{dn}{\;u}.\end{array}$
(220)

As in the case of Lamé equation [33], these properties give four types of
solutions for the associated Lamé equation: even and odd solutions with period
$2K$, even and odd solutions with period $4K$. In Sections 5.1 and 5.2,
solutions of each type are put together. We also take into account the special
values

$\displaystyle\begin{array}[]{l}\mathrm{sn}\;0=0,\quad\mathrm{sn}\;K=1,\quad\mathrm{cn}\;0=1,\quad\mathrm{cn}{\;K}=0,\quad\mathrm{dn}\;0=1,\quad\mathrm{dn}{\;K}=1-k^{2}.\end{array}$
(222)

###  5.1 Finite-series eigenfunctions
$\left(\mathring{\psi}^{(i)},\tilde{\psi}^{(i)}\right)$ in terms of
$\mathrm{sn}^{2}u$

For the associated Lamé equation (202), the power series solutions
$\mathring{{H}}^{(i)}(x)$ lead to eight pairs of finite-series expansions
$\left(\mathring{\psi}^{(i)},\tilde{\psi}^{(i)}\right)$ by means of the
truncation procedure indicated in (216). To write down the four types of
expansions we observe that the pairs
$\left(\mathring{\psi}^{(j+4)},\tilde{\psi}^{(j+4)}\right)$ ($j=1,2,3,4$) can
be generated from $\left(\mathring{\psi}^{(j)},\tilde{\psi}^{(j)}\right)$ by
the substitutions $\bm{l}\mapsto-\bm{l}-1$ and $\bm{m}\mapsto-\bm{m}-1$ which
preserve the period and parity of the expansions.

The series coefficients for
$\left(\mathring{\psi}^{(i)},\tilde{\psi}^{(i)}\right)$ are denoted by
$\left(\mathring{b}_{n}^{(i)},\tilde{b}_{n}^{(i)}\right)$. However, in the
recurrence relations
$\mathring{\alpha}_{n}\mathring{b}_{n+1}+\mathring{\beta}_{n}\mathring{b}_{n}+\mathring{\gamma}_{n}\mathring{b}_{n-1}=0$
and
$\tilde{\alpha}_{n}\tilde{b}_{n+1}+\tilde{\beta}_{n}\tilde{b}_{n}+\tilde{\gamma}_{n}\tilde{b}_{n-1}=0$
, the coefficients are formally identical, that is,

$\displaystyle\tilde{{\alpha}}_{n}^{(i)}=\mathring{{\alpha}}_{n}^{(i)},\qquad\tilde{{\beta}}_{n}^{(i)}=\mathring{{\beta}}_{n}^{(i)},\qquad\tilde{{\gamma}}_{n}^{(i)}=\mathring{{\gamma}}_{n}^{(i)}.$
(223)

On the right-hand side of each expansion, inside square brackets, we write the
restrictions on the parameters $\bm{l}$ and $\bm{m}$ that assure the Arscott
condition (82).

First we write the four types of expansions, then we discuss the Arscott
condition (82) and, finally, we exhibit some problems which can be solved by
these expansions.

####  5.1.1 First type: Even solutions, period $2K$

$\displaystyle\begin{cases}\mathring{{\psi}}^{(1)}(u)=\mathrm{dn}^{\bm{l}+1}u\;\displaystyle\sum_{n=0}^{\frac{\bm{m-l}-1}{2}}\mathring{{b}}_{n}^{(1)}\;\mathrm{sn}^{2n}u\quad\mbox{
if}\quad\bm{m}-\bm{l}=1,3,5,\cdots,\;\\\ \hskip
227.62204pt\left[\;\bm{m}\leq-\frac{1}{2},\;\bm{m}+\bm{l}<-2\;\right],\vspace{3mm}\\\
\tilde{{\psi}}^{(1)}(u)=\mathrm{dn}^{\bm{l}+1}u\;\displaystyle\sum_{n=0}^{-\frac{\bm{l+m}+2}{2}}\tilde{{b}}_{n}^{(1)}\quad\mathrm{sn}^{2n}u\;\mbox{
if }\quad\bm{m}+\bm{l}=-2,-4,-6,\cdots,\;\\\ \hskip
227.62204pt\left[\bm{m}\geq-\frac{1}{2},\;\bm{m}-\bm{l}>1\;\right],\end{cases}$
(224a)
$\displaystyle\begin{array}[]{l}\mathring{{\alpha}}_{n}^{(1)}=\frac{1}{k^{2}}\left(n+\frac{1}{2}\right)(n+1),\qquad\mathring{{\beta}}_{n}^{(1)}=\left(\frac{1}{k^{2}}+1\right)n^{2}-(\bm{l}+1)n+\end{array}\vspace{2mm}\vspace{3mm}$
(224c) $\displaystyle\begin{array}[]{l}\frac{1}{4k^{2}}{\cal
E}-\frac{1}{4}(\bm{l}+1)^{2},\quad\mathring{{\gamma}}_{n}^{(1)}=\left(n-\frac{\bm{m}-\bm{l}+1}{2}\right)\left(n+\frac{\bm{l}+\bm{m}}{2}\right);\end{array}$
(224e)

Only one of the above expansions is valid when (from now on, $M_{i}$ and
$N_{i}$ denote positive integers)

$\displaystyle\begin{array}[]{l}\bm{m-l}-1=2M_{1}\;\mbox{ and
}\;\bm{m+l}+2=-2N_{1},\quad\left(\Leftrightarrow\bm{m}=M_{1}-N_{1}-\frac{1}{2},\;\right.\end{array}\vspace{2mm}$
(224g)
$\displaystyle\begin{array}[]{l}\left.\bm{l}=-M_{1}-N_{1}-\frac{3}{2}\right):\mathring{\psi}^{(1)}\mbox{
if }\;M_{1}<N_{1},\quad\tilde{\psi}^{(1)}\mbox{ if }N_{1}<M_{1}.\end{array}$
(224i)

For series with two or more terms ($M_{1}\geq 1$ or $N_{1}\geq 1$), we see
that these expansions satisfy the Arscott criterion. The same holds for the
other seven cases given below: (225h), (226i),$\cdots$, (231g).

$\displaystyle\begin{cases}\mathring{{\psi}}^{(5)}(u)=\mathrm{dn}^{-\bm{l}}u\;\displaystyle\sum_{n=0}^{\frac{\bm{l-m}-1}{2}}\mathring{{b}}_{n}^{(5)}\;\mathrm{sn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{l}-\bm{m}=1,3,5,\cdots,\end{array}\\\ \hskip
227.62204pt\begin{array}[]{l}\left[\;\bm{m}\geq-\frac{1}{2},\;\bm{m}+\bm{l}>0\;\right],\end{array}\vspace{3mm}\\\
\tilde{{\psi}}^{(5)}(u)=\mathrm{dn}^{-\bm{l}}u\;\displaystyle\sum_{n=0}^{\frac{\bm{l+m}}{2}}\;\tilde{{b}}_{n}^{(5)}\;\mathrm{sn}^{2n}u\qquad\mbox{
if }\quad\begin{array}[]{l}\bm{m}+\bm{l}=0,2,4,\cdots,\end{array}\qquad\\\
\begin{array}[]{l}\hskip
227.62204pt\left[\bm{m}\leq-\frac{1}{2},\;\bm{l}-\bm{m}>1\;\right],\end{array}\end{cases}$
(225a)
$\displaystyle\begin{array}[]{l}\mathring{{\alpha}}_{n}^{(5)}=\frac{1}{k^{2}}\left(n+\frac{1}{2}\right)(n+1),\end{array}\begin{array}[]{l}\mathring{{\beta}}_{n}^{(5)}=-\left(\frac{1}{k^{2}}+1\right)n^{2}+\bm{l}n+\frac{1}{4k^{2}}{\cal
E}-\frac{1}{4}\bm{l}^{2},\end{array}\quad\vspace{2mm}\vspace{3mm}$ (225f)
$\displaystyle\begin{array}[]{l}\mathring{{\gamma}}_{n}^{(5)}=\left(n-\frac{\bm{l}-\bm{m}+1}{2}\right)\left(n-\frac{\bm{l}+\bm{m}+2}{2}\right).\end{array}$

Only one expansion when

$\displaystyle\begin{array}[]{l}\bm{l-m}-1=2M_{2}\;\mbox{ and
}\;\bm{m+l}=2N_{2},\quad\left(\Leftrightarrow\bm{m}=N_{2}-M_{2}-\frac{1}{2},\;\right.\end{array}\vspace{2mm}$
(225h)
$\displaystyle\begin{array}[]{l}\left.\bm{l}=-M_{2}+N_{2}+\frac{1}{2}\right):\mathring{\psi}^{(5)}\mbox{
if }\;M_{2}<N_{2},\quad\tilde{\psi}^{(5)}\mbox{ if }\;N_{2}<M_{2}.\end{array}$
(225j)

####  5.1.2 Second type: Odd solutions, period $4K$

$\displaystyle\begin{cases}\mathring{{\psi}}^{(2)}(u)=\mathrm{sn}u\;\mathrm{dn}^{\bm{l}+1}u\displaystyle\sum_{n=0}^{\frac{\bm{m-l}-2}{2}}\mathring{{b}}_{n}^{(2)}\;\mathrm{sn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}-\bm{l}=2,4,6,\cdots,\end{array}\\\
\begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}\leq-\frac{1}{2},\;\;\bm{m}+\bm{l}<-3\;\right]\end{array},\vspace{3mm}\\\
\tilde{{\psi}}^{(2)}(u)=\mathrm{sn}u\;\mathrm{dn}^{\bm{l}+1}u\displaystyle\sum_{n=0}^{-\frac{\bm{m+l}+3}{2}}\tilde{{b}}_{n}^{(2)}\;\mathrm{sn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}+\bm{l}=-3,-5,-7,\cdots,\end{array}\\\
\begin{array}[]{l}\hskip
227.62204pt\;\;\left[\bm{m}\geq-\frac{1}{2},\;\;\bm{m}-\bm{l}>2\;\right],\end{array}\end{cases}$
(226a)
$\displaystyle\begin{array}[]{l}\mathring{{\alpha}}_{n}^{(2)}=\frac{1}{k^{2}}\left(n+\frac{3}{2}\right)(n+1),\quad\mathring{{\beta}}_{n}^{(2)}=-\left(\frac{1}{k^{2}}+1\right)n^{2}-\left(\frac{1}{k^{2}}+\bm{l}+2\right)n+\end{array}\vspace{3mm}$
(226c) $\displaystyle\begin{array}[]{l}\frac{1}{4k^{2}}({\cal
E}-1)-\frac{1}{4}(\bm{l}+2)^{2},\quad\mathring{{\gamma}}_{n}^{(2)}=\left(n-\frac{\bm{m}-\bm{l}}{2}\right)\left(n+\frac{\bm{l}+\bm{m}+1}{2}\right);\end{array},$
(226e)

Only one expansion when

$\displaystyle\begin{array}[]{l}\bm{m-l}-2=2M_{3}\;\mbox{ and
}\;\bm{m+l}+3=-2N_{3},\quad\left(\Leftrightarrow\bm{m}=M_{3}-N_{3}-\frac{1}{2},\right.\end{array}\vspace{2mm}$
(226i)
$\displaystyle\begin{array}[]{l}\left.\bm{l}=-M_{3}-N_{3}-\frac{5}{2}\right):\mathring{\psi}^{(2)}\mbox{
if }\;M_{3}<N_{3},\;\tilde{\psi}^{(2)}\mbox{ if }\;N_{3}<M_{3}.\end{array}$
$\displaystyle\begin{cases}\mathring{{\psi}}^{(6)}(u)=\mathrm{sn}u\;\mathrm{dn}^{-\bm{l}}u\displaystyle\sum_{n=0}^{\frac{\bm{l-m}-2}{2}}\mathring{{b}}_{n}^{(6)}\;\mathrm{sn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{l}-\bm{m}=2,4,6,\cdots,\end{array}\\\
\begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}\geq-\frac{1}{2},\;\;\bm{m}+\bm{l}>1\;\right]\end{array},\vspace{3mm}\\\
\tilde{{\psi}}^{(6)}(u)=\mathrm{sn}u\;\mathrm{dn}^{-\bm{l}}u\displaystyle\sum_{n=0}^{\frac{\bm{m+l}-1}{2}}\tilde{{b}}_{n}^{(6)}\;\mathrm{sn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}+\bm{l}=1,3,5,\cdots,\end{array}\\\
\begin{array}[]{l}\hskip
227.62204pt\;\;\left[\bm{m}\leq-\frac{1}{2},\;\;\bm{l}-\bm{m}>2\;\right],\end{array}\end{cases}$
(227a)
$\displaystyle\begin{array}[]{l}\mathring{{\alpha}}_{n}^{(6)}=\frac{1}{k^{2}}\left(n+\frac{3}{2}\right)(n+1),\quad\mathring{{\beta}}_{n}^{(6)}=-\left(\frac{1}{k^{2}}+1\right)n^{2}-\left(\frac{1}{k^{2}}-\bm{l}+1\right)n+\end{array}\vspace{3mm}$
(227c) $\displaystyle\begin{array}[]{l}\frac{1}{4k^{2}}({\cal
E}-1)-\frac{1}{4}(\bm{l}-1)^{2},\qquad\mathring{{\gamma}}_{n}^{(6)}=\left(n-\frac{\bm{l}-\bm{m}}{2}\right)\left(n-\frac{\bm{l}+\bm{m}+1}{2}\right)\end{array},$
(227e)

Only one expansion when

$\displaystyle\begin{array}[]{l}\bm{l-m}-2=2M_{4}\;\mbox{ and
}\;\bm{m+l}-1=2N_{4}\quad\left(\Leftrightarrow\bm{m}=N_{4}-M_{4}-\frac{1}{2},\right.\end{array}\vspace{2mm}$
(227g)
$\displaystyle\begin{array}[]{l}\left.\bm{l}=M_{4}+N_{4}+\frac{3}{2}\right):\mathring{\psi}^{(6)}\mbox{
if }\;M_{4}<N_{4},\quad\tilde{\psi}^{(6)}\mbox{ if }\;N_{4}<M_{4}.\end{array}$
(227i)

####  5.1.3 Third type: Even solutions, period $4K$

$\displaystyle\begin{cases}\mathring{{\psi}}^{(3)}(u)=\mathrm{cn}u\;\mathrm{dn}^{\bm{l}+1}u\;\displaystyle\sum_{n=0}^{\frac{\bm{m-l}-2}{2}}\mathring{{b}}_{n}^{(3)}\;\mathrm{sn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}-\bm{l}=2,4,6,\cdots,\end{array}\\\
\begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}\leq-\frac{1}{2},\;\;\bm{m}+\bm{l}<-3\;\right]\end{array}\vspace{3mm}\\\
\tilde{{\psi}}^{(3)}(u)=\mathrm{cn}u\;\mathrm{dn}^{\bm{l}+1}u\;\displaystyle\sum_{n=0}^{-\frac{\bm{m+l}+3}{2}}\tilde{{b}}_{n}^{(3)}\;\mathrm{sn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}+\bm{l}=-3,-5,-7,\cdots,\quad;\end{array}\\\
\begin{array}[]{l}\hskip 227.62204pt\left[\bm{m}\geq-\frac{1}{2},\;\
\bm{m}-\bm{l}>2\;\right],\end{array}\end{cases}$ (228a)
$\displaystyle\begin{array}[]{l}\mathring{{\alpha}}_{n}^{(3)}=\frac{1}{k^{2}}\left(n+\frac{3}{2}\right)(n+1),\qquad\mathring{{\beta}}_{n}^{(3)}=-\left(\frac{1}{k^{2}}+1\right)n^{2}-\left(\frac{1}{k^{2}}+\bm{l}+1\right)n+\end{array}\vspace{2mm}$
(228c) $\displaystyle\begin{array}[]{l}\frac{1}{4k^{2}}({\cal
E}-1)-\frac{1}{4}(\bm{l}+1)^{2},\qquad\mathring{\gamma}_{n}^{(3)}=\mathring{\gamma}_{n}^{(2)}.\end{array}$
(228e)

Only one expansion when

$\displaystyle\begin{array}[]{l}\bm{m-l}-2=2M_{3}\;\mbox{ and
}\;\bm{m+l}+3=-2N_{3}\quad\left(\Leftrightarrow\bm{m}=M_{3}-N_{3}-\frac{1}{2},\right.\end{array}\vspace{2mm}$
(228g)
$\displaystyle\begin{array}[]{l}\left.\bm{l}=-M_{3}-N_{3}-\frac{5}{2}\right):\mathring{\psi}^{(3)}\mbox{
if }\;M_{3}<N_{3},\quad\tilde{\psi}^{(3)}\mbox{ if }\;N_{3}<M_{3}.\end{array}$
(228i)
$\displaystyle\begin{cases}\mathring{{\psi}}^{(7)}(u)=\mathrm{cn}u\;\mathrm{dn}^{-\bm{l}}u\;\displaystyle\sum_{n=0}^{\frac{\bm{l-m}-2}{2}}\mathring{{b}}_{n}^{(7)}\;\mathrm{sn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{l}-\bm{m}=2,4,6,\cdots,\hskip
19.91692pt;\end{array}\\\ \begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}\geq-\frac{1}{2},\;\;\bm{m}+\bm{l}>1\;\right]\end{array},\vspace{3mm}\\\
\tilde{{\psi}}^{(7)}(u)=\mathrm{cn}u\;\mathrm{dn}^{-\bm{l}}u\;\displaystyle\sum_{n=0}^{\frac{\bm{m+l}-1}{2}}\tilde{{b}}_{n}^{(7)}\;\mathrm{sn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}+\bm{l}=1,3,5,\cdots,\qquad;\end{array}\\\
\begin{array}[]{l}\hskip
227.62204pt\left[\bm{m}\leq-\frac{1}{2},\;\;\bm{l}-\bm{m}>2\;\right],\end{array}\end{cases}$
(229a)
$\displaystyle\begin{array}[]{l}\mathring{{\alpha}}_{n}^{(7)}=\frac{1}{k^{2}}\left(n+\frac{3}{2}\right)(n+1),\qquad\mathring{{\beta}}_{n}^{(7)}=-\left(\frac{1}{k^{2}}+1\right)n^{2}-\left(\frac{1}{k^{2}}-\bm{l}\right)n+\end{array}\vspace{3mm}$
(229c) $\displaystyle\begin{array}[]{l}\frac{1}{4k^{2}}({\cal
E}-1)-\frac{1}{4}\bm{l}^{2},\qquad\mathring{\gamma}_{n}^{(7)}=\mathring{\gamma}_{n}^{(6)}.\end{array}$
(229e)

Only one expansion when

$\displaystyle\begin{array}[]{l}\bm{l-m}-2=2M_{4}\;\mbox{ and
}\;\bm{m+l}-1=2N_{4}\quad\left(\Leftrightarrow\bm{m}=N_{4}-M_{4}-\frac{1}{2},\right.\end{array}\vspace{2mm}$
(229g)
$\displaystyle\begin{array}[]{l}\left.\bm{l}=M_{2}+N_{4}+\frac{3}{2}\right):\mathring{\psi}^{(7)}\mbox{
if }\;M_{4}<N_{4},\quad\tilde{\psi}^{(7)}\mbox{ if }\;N_{4}<M_{4}.\end{array}$
(229i)

####  5.1.4 Fourth type: Odd solutions, period $2K$

$\displaystyle\begin{cases}\mathring{{\psi}}^{(4)}(u)=\mathrm{sn}u\;\mathrm{cn}u\;\mathrm{dn}^{\bm{l}+1}u\displaystyle\sum_{n=0}^{\frac{\bm{m-l}-3}{2}}\mathring{{b}}_{n}^{(4)}\;\mathrm{sn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}-\bm{l}=3,5,7,\cdots,\hskip
36.98866pt;\end{array}\\\ \begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}\leq-\frac{1}{2},\;\;\bm{m}+\bm{l}<-4\;\right]\end{array}\vspace{3mm}\\\
\tilde{{\psi}}^{(4)}(u)=\mathrm{sn}u\;\mathrm{cn}u\;\mathrm{dn}^{\bm{l}+1}u\displaystyle\sum_{n=0}^{-\frac{\bm{m+l}+4}{2}}\tilde{{b}}_{n}^{(4)}\;\mathrm{sn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}+\bm{l}=-4,-6,-8,\cdots,\;;\end{array}\\\
\begin{array}[]{l}\hskip
227.62204pt\left[\bm{m}\geq-\frac{1}{2},\;\;\bm{m}-\bm{l}>3\;\right]\end{array}\end{cases}$
(230a)
$\displaystyle\begin{array}[]{l}\mathring{{\alpha}}_{n}^{(4)}=\frac{1}{k^{2}}\left(n+\frac{3}{2}\right)(n+1),\qquad\mathring{{\beta}}_{n}^{(4)}=-\left(\frac{1}{k^{2}}+1\right)n^{2}-\left(\frac{2}{k^{2}}+\bm{l}+2\right)n+\end{array}\vspace{2mm}$
(230c) $\displaystyle\begin{array}[]{l}\frac{1}{4k^{2}}({\cal
E}-4)-\frac{1}{4}(\bm{l}+2)^{2},\qquad\mathring{\bm{\gamma}}_{n}^{(4)}=\left(n-\frac{\bm{m}-\bm{l}-1}{2}\right)\left(n+\frac{\bm{l}+\bm{m}+2}{2}\right);\end{array}$
(230e)

Only one expansion when

$\displaystyle\begin{array}[]{l}\bm{m-l}-3=2M_{5}\;\mbox{ and
}\;\bm{m+l}+4=-2N_{5},\quad\left(\Leftrightarrow\bm{m}=M_{5}-N_{5}-\frac{1}{2},\right.\end{array}\vspace{2mm}$
(230g)
$\displaystyle\begin{array}[]{l}\left.\bm{l}=-M_{5}-N_{5}-\frac{3}{2}\right):\mathring{\psi}^{(4)}\mbox{
if }\;M_{5}<N_{5},\quad\tilde{\psi}^{(4)}\mbox{ if }\;N_{5}<M_{5}.\end{array}$
(230i)
$\displaystyle\begin{cases}\mathring{{\psi}}^{(8)}(u)=\mathrm{sn}u\;\mathrm{cn}u\;\mathrm{dn}^{-\bm{l}}u\displaystyle\sum_{n=0}^{\frac{\bm{l-m}-3}{2}}\mathring{{b}}_{n}^{(8)}\;\mathrm{sn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{l}-\bm{m}=3,5,7,\cdots,\hskip
19.91692pt;\end{array}\\\ \begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}\geq-\frac{1}{2},\;\;\bm{m}+\bm{l}>2\;\right],\end{array}\vspace{3mm}\\\
\tilde{{\psi}}^{(8)}(u)=\mathrm{sn}u\;\mathrm{cn}u\;\mathrm{dn}^{-\bm{l}}u\displaystyle\sum_{n=0}^{\frac{\bm{m+l}-2}{2}}\tilde{{b}}_{n}^{(8)}\;\mathrm{sn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}+\bm{l}=2,4,6,\cdots,;\end{array}\\\
\begin{array}[]{l}\hskip
227.62204pt\qquad\left[\bm{m}\leq-\frac{1}{2},\;\;\bm{l}-\bm{m}>3\;\right],\end{array}\end{cases}$
(231a)
$\displaystyle\begin{array}[]{l}\mathring{{\alpha}}_{n}^{(8)}=\frac{1}{k^{2}}\left(n+\frac{3}{2}\right)(n+1),\qquad\mathring{{\beta}}_{n}^{(5)}=-\left(\frac{1}{k^{2}}+1\right)n^{2}-\left(\frac{2}{k^{2}}-\bm{l}+1\right)n+\end{array}\vspace{2mm}$
(231c) $\displaystyle\begin{array}[]{l}\frac{1}{4k^{2}}({\cal
E}-4)-\frac{1}{4}(\bm{l}-1)^{2},\qquad\mathring{\bm{\gamma}}_{n}^{(8)}=\left(n-\frac{\bm{l}-\bm{m}-1}{2}\right)\left(n-\frac{\bm{l}+\bm{m}}{2}\right).\end{array}$
(231e)

Only one expansion when

$\displaystyle\begin{array}[]{l}\bm{l-m}-3=2M_{6}\;\mbox{ and
}\;\bm{m+l}-2=-2N_{6}\quad\left(\Leftrightarrow\bm{m}=N_{6}-M_{6}-\frac{1}{2},\right.\end{array}\vspace{2mm}$
(231g)
$\displaystyle\begin{array}[]{l}\left.\bm{l}=M_{6}+N_{6}+\frac{5}{2}\right):\mathring{\psi}^{(8)}\mbox{
if }\;M_{6}<N_{6},\quad\tilde{\psi}^{(8)}\mbox{ if }\;N_{6}<M_{6}.\end{array}$
(231i)

Due to the symmetry under the substitutions (206), we discuss the Arscott
condition (82) only for
$\left(\mathring{\psi}^{(i)},\tilde{\psi}^{(i)}\right)$ with $i=1,2,3,4$ . As
the $\mathring{{\alpha}}_{n}^{(i)}$ are positive, the condition
$\mathring{\alpha}_{n-1}\mathring{\gamma}_{n}>0$ is satisfied if
$\mathring{\gamma}_{n}>0$ for any admissible $n$. So, for
$\mathring{{\psi}}^{(i)}$ we requires that

$\displaystyle\begin{array}[]{r}\mathring{{\gamma}}_{n}^{(1)}=\left(n-\frac{\bm{m}-\bm{l}+1}{2}\right)\left(n+\frac{\bm{l}+\bm{m}}{2}\right)>0\;\mbox{for}\;1\leq
n\leq\frac{\bm{m}-\bm{l}-1}{2},\;\left[\bm{m}-\bm{l}=3,5,7,\cdots\right],\end{array}$
$\displaystyle\begin{array}[]{r}\mathring{{\gamma}}_{n}^{(2)}=\mathring{\bm{\gamma}}_{n}^{(3)}=\left(n-\frac{\bm{m}-\bm{l}}{2}\right)\left(n+\frac{\bm{l}+\bm{m}+1}{2}\right)>0\;\mbox{
for }\;1\leq
n\leq\frac{\bm{m}-\bm{l}-2}{2},\;\left[{m}-\bm{l}=4,6,8,\cdots\right],\end{array}$
$\displaystyle\begin{array}[]{r}\mathring{\bm{\gamma}}_{n}^{(4)}=\left(n-\frac{\bm{m}-\bm{l}-1}{2}\right)\left(n+\frac{\bm{l}+\bm{m}+2}{2}\right)>0\;\mbox{for}\;1\leq
n\leq\frac{\bm{m}-\bm{l}-3}{2},\;\left[\bm{m}-\bm{l}=5,7,9,\cdots\right].\end{array}$

Thus, since the first term is negative, $\mathring{\gamma}_{n}>0$ if the last
term on the right of the above equations is also negative for each value of
$n$. By considering the maximum value of $n$ we find that $\bm{m}<1/2$ for the
four expansions; to include all values of $n$, we take $\bm{m}\leq-1/2$ due to
(237) and similar relations for each $\mathring{\gamma}_{n}^{(i)}$ . On the
other side, for series with several terms, $\mathring{\gamma}_{n}^{(i)}>0$
implies $\mathring{\gamma}_{1}^{(i)}>0$: this imposes the following
restrictions on $\bm{m}+\bm{l}$

$\displaystyle\begin{array}[]{l}\bm{m}\leq-\frac{1}{2}\mbox{ for
}\mathring{{\psi}}^{(i)},\quad\bm{m}+\bm{l}+2<0\;\mbox{ for
}\;\mathring{{\psi}}^{(1)},\;\;\bm{m}+\bm{l}+3<0\;\mbox{ for
}\;\mathring{{\psi}}^{(2)}\mbox{ and
}\mathring{{\psi}}^{(3)},\end{array}\vspace{3mm}$ (233)
$\displaystyle\begin{array}[]{l}\bm{l}+\bm{m}+4<0\;\mbox{ for
}\;\mathring{{\psi}}^{(4)}\end{array},$ (235)

($i=1,2,3,4$) in addition to the restrictions on $\bm{m}-\bm{l}$. Actually,
firstly we find conditions to get $\mathring{\gamma}_{n}^{(i)}\leq 0$ by
considering $n$ as a continuous variable. For example,

$\displaystyle\begin{array}[]{l}\mathring{\bm{\gamma}}_{n}^{(1)}\leq 0\;\mbox{
if }\begin{cases}\frac{\bm{m}-\bm{l}+1}{2}\leq
n\leq-\frac{\bm{m}+\bm{l}}{2}\qquad\mbox{ when
}\qquad\bm{m}\leq-\frac{1}{2},\vspace{2mm}\\\ -\frac{\bm{m}+\bm{l}}{2}\leq
n\leq\frac{\bm{m}-\bm{l}+1}{2}\qquad\mbox{ when
}\qquad\bm{m}\geq-\frac{1}{2}.\end{cases}\end{array}$ (237)

Thence, $\mathring{\gamma}_{n}^{(i)}>0$ when $n$ lies outside the above
intervals. In particular,

$\displaystyle\begin{array}[]{l}\mathring{\bm{\gamma}}_{n}^{(1)}>0\;\mbox{ for
}n<\frac{\bm{m}-\bm{l}+1}{2},\mbox{ with
}\;\bm{m}\leq-\frac{1}{2},\end{array}$

an interval which includes the subinterval $n\leq(\bm{m}-\bm{l}-1)/2$.

On the other side, for $\tilde{{\psi}}^{(i)}$, we demand that
($\mathring{{\gamma}}_{n}^{(2)}=\mathring{{\gamma}}_{n}^{(3)}$)

$\displaystyle\begin{array}[]{r}\mathring{{\gamma}}_{n}^{(1)}=\left(n+\frac{\bm{l}+\bm{m}}{2}\right)\left(n+\frac{\bm{l}-\bm{m}-1}{2}\right)>0\;\mbox{for}\;1\leq
n\leq-\frac{\bm{m}+\bm{l}+2}{2},\;\left[\bm{m}+\bm{l}=-4,-6,-8,\cdots\right],\end{array}$
$\displaystyle\begin{array}[]{r}\mathring{{\gamma}}_{n}^{(2)}=\left(n+\frac{\bm{l}+\bm{m}+1}{2}\right)\left(n+\frac{\bm{l}-\bm{m}}{2}\right)>0\;\mbox{for}\;1\leq
n\leq-\frac{\bm{m}+\bm{l}+3}{2},\;\left[{m}+\bm{l}=-5,-7,-9,\cdots\right],\end{array}$
$\displaystyle\begin{array}[]{r}\mathring{{\gamma}}_{n}^{(4)}=\left(n+\frac{\bm{l}+\bm{m}+2}{2}\right)\left(n+\frac{\bm{l}-\bm{m}+1}{2}\right)>0\;\mbox{for}\;1\leq
n\leq-\frac{\bm{m}+\bm{l}+4}{2},\;\left[\bm{m}-\bm{l}=-6,-8,\cdots\right].\end{array}$

As the first term is negative, then $\mathring{\gamma}_{n}^{(i)}>0$ if the
second term on the right of these equations is also negative for each value of
$n$. From (237) and similar relations for each $\mathring{\gamma}_{n}^{(i)}$,
we find that $\bm{m}\geq-{1}/{2}$. Besides this,
$\mathring{\gamma}_{1}^{(i)}>0$ restricts the values of $\bm{m}-\bm{l}$,
namely,

$\displaystyle\begin{array}[]{l}\bm{m}\geq-\frac{1}{2}\mbox{ for
}\tilde{{\psi}}^{(i)},\quad\bm{l}-\bm{m}+1<0\;\mbox{ for
}\;\tilde{{\psi}}^{(1)},\;\;\bm{l}-\bm{m}+2<0\;\mbox{ for
}\;\tilde{{\psi}}^{(2)}\mbox{ and
}\tilde{{\psi}}^{(3)},\end{array}\vspace{3mm}$ (240)
$\displaystyle\begin{array}[]{l}\bm{l}-\bm{m}+3<0\;\mbox{ for
}\;\tilde{{\psi}}^{(4)},\end{array}$ (242)

($i=1,2,3,4$) in addition to the restrictions on $\bm{m}+\bm{l}$. Using (237)
we can check, for example, that

$\displaystyle\begin{array}[]{l}\mathring{\bm{\gamma}}_{n}^{(1)}>0\;\mbox{ for
}n<-\frac{\bm{l}+\bm{m}}{2},\mbox{ when }\;\bm{m}\geq-\frac{1}{2},\end{array}$

where the above interval includes all the possible values for $n$, that is,
$n\leq-(\bm{l}+\bm{m}+2)/2$.

To proceed we mention some examples, Obviously, the cases (224g), (226i),
$\cdots$, (231g) are solved by finite series of $\mathrm{sn}^{2}u$.
Furthermore, we find that each of these cases admits a second expansion having
the same period and opposite parity. For instance, if $0<M_{1}<N_{1}$ in
(224g), we have

$\displaystyle\begin{array}[]{l}\mbox{for }\bm{m}=M_{1}-N_{1}-\frac{1}{2}\\\
\mbox{and
}\bm{l}=-M_{1}-N_{1}-\frac{3}{2}\end{array}:\begin{cases}\mathring{\psi}^{(1)}\mbox{
with }0\leq n\leq M_{1}<N_{1}\mbox{, even, period }2K;\vspace{2mm}\\\
\mathring{\psi}^{(4)}\mbox{ with }0\leq n\leq(M_{1}-1),\mbox{ odd, period
}2K;\end{cases}$ (246)

and so far on. In each case there are only two distinct expansions satisfying
the Arscott condition.

As a second example we take $\bm{m}=-\frac{1}{2}$ in which case
$\mathring{{\psi}}^{(i)}(u)=\tilde{{\psi}}^{(i)}(u)$. The eight expansions
satisfy the Arscott condition with the following values for $\bm{l}$:

$\displaystyle\begin{array}[]{l}\mbox{even solutions, period
}2K\;(\bm{m}=-1/2):\left\\{\begin{array}[]{l}\mathring{\psi}^{(1)}\;\mbox{ if
}\;\bm{l}=-\frac{3}{2},-\frac{7}{2},-\frac{11}{2},\cdots;\vspace{3mm}\\\
\mathring{{\psi}}^{(5)}\;\mbox{ if
}\;\bm{l}=\frac{1}{2},\frac{5}{2},\frac{9}{2},\cdots;\end{array}\right.\end{array}\vspace{3mm}$
(250) $\displaystyle\begin{array}[]{l}\mbox{odd solutions, period
}2K\;(\bm{m}=-1/2):\left\\{\begin{array}[]{l}\mathring{\psi}^{(4)}\;\mbox{ if
}\;\bm{l}=-\frac{7}{2},-\frac{11}{2},-\frac{15}{2}\cdots;\vspace{2mm}\\\
\mathring{{\psi}}^{(8)}\;\mbox{ if
}\;\bm{l}=\frac{5}{2},\frac{9}{2},\frac{13}{2}\cdots;\end{array}\right.\vspace{2mm}\end{array}$
(254) $\displaystyle\begin{array}[]{l}\mbox{odd solutions, period
}4K\;(\bm{m}=-1/2):\left\\{\begin{array}[]{l}\mathring{\psi}^{(2)}\;\mbox{ if
}\;\bm{l}=-\frac{5}{2},-\frac{9}{2},-\frac{13}{2}\cdots;\\\
\mathring{{\psi}}^{(6)}\;\mbox{ if
}\;\bm{l}=\frac{3}{2},\frac{7}{2},\frac{11}{2}\cdots;\end{array}\right.\vspace{2mm}\end{array}$
(258) $\displaystyle\begin{array}[]{l}\mbox{even solutions, period
}4K\;(\bm{m}=-1/2):\left\\{\begin{array}[]{l}\mathring{\psi}^{(3)}\;\mbox{ if
}\;\bm{l}=-\frac{5}{2},-\frac{9}{2},-\frac{13}{2}\cdots;\vspace{2mm}\\\
\mathring{{\psi}}^{(7)}\;\mbox{ if
}\;\bm{l}=\frac{3}{2},\frac{7}{2},\frac{11}{2},\cdots,\end{array}\right.\end{array}$
(262)

In the above expressions, there are at the most two distinct expansions for a
given value of $\bm{l}$ (only one solution when $\bm{l}=-3/2$ or
$\bm{l}=1/2$). Notice that the Arscott condition is applicable for series with
two or more terms; further, if the condition is not fulfilled, we cannot in
advance state anything on the eigenvalues of the characteristic equation.

As a last illustration, we consider solutions for Eq. (211), that is, for
$\bm{m}=0$ or $\bm{m}=-1$. There are finite-series solutions in series of
$\mathrm{sn}^{2n}u$ for any integer value of $\bm{l}$ (excepting $\bm{l}=-1$),
but only the following eight expansions satisfy the Asrcott condition

$\displaystyle\begin{array}[]{l}\mbox{even solutions, period
}2K\;(\bm{m}=0):\left\\{\begin{array}[]{l}\tilde{\psi}^{(1)}\quad\mbox{ if
}\quad\bm{l}=-2,-4,-6,\cdots;\vspace{2mm}\\\
\mathring{{\psi}}^{(5)}\quad\mbox{ if
}\quad\bm{l}=1,3,5,\cdots;\end{array}\right.\end{array}$ (266)
$\displaystyle\begin{array}[]{l}\mbox{odd solutions, period
}2K\;(\bm{m}=0):\left\\{\begin{array}[]{l}\tilde{\psi}^{(4)}\quad\mbox{ if
}\quad\bm{l}=-4,-6-8,\cdots;\vspace{2mm}\\\ \mathring{{\psi}}^{(8)}\quad\mbox{
if }\quad\bm{l}=3,5,7,\cdots;\end{array}\right.\end{array}$ (270)
$\displaystyle\begin{array}[]{l}\mbox{odd solutions, period
}4K\;(\bm{m}=0):\left\\{\begin{array}[]{l}\tilde{\psi}^{(2)}\quad\mbox{ if
}\quad\bm{l}=-3,-5,-7,\cdots;\vspace{2mm}\\\
\mathring{{\psi}}^{(6)}\quad\mbox{ if
}\quad\bm{l}=-2,-4,-6,\cdots;\end{array}\right.\end{array}$ (274)
$\displaystyle\begin{array}[]{l}\mbox{even solutions, period
}4K\;(\bm{m}=0):\left\\{\begin{array}[]{l}\tilde{\psi}^{(3)}\quad\mbox{ if
}\quad\bm{l}=-3,-5,-7,\cdots;\vspace{2mm}\\\
\mathring{{\psi}}^{(7)}\quad\mbox{ if
}\quad\bm{l}=-2,-4,-6,\cdots.\end{array}\right.\end{array}$ (278)

On the other hand, for $\bm{m}=-1$ the remaining expansions also lead to
solutions satisfying the Arscott condition; nevertheless, such solutions are
identical to the above ones. For example, we find that

$\displaystyle\begin{array}[]{l}\mathring{\psi}^{(1)}\big{|}_{\bm{m}=-1}=\tilde{\psi}^{(1)}\big{|}_{\bm{m}=0},\qquad\tilde{\psi}^{(5)}\big{|}_{\bm{m}=-1}=\mathring{\psi}^{(5)}\big{|}_{\bm{m}=0},\qquad\mbox{and
so on.}\end{array}$

From relations (266-278), we find that there are four types of finite-series
solutions if $\bm{l}=-4,-6,-8,\cdots$, unlike the expansions of the previous
examples.

###  5.2 Finite-series eigenfunction
$\left(\mathring{\bm{\Psi}}^{(i)},\tilde{\bm{\Psi}}^{(i)}\right)$ in terms of
$\mathrm{cn}^{2}u$

This is similar to Section 5.1. The finite-series expansions
$\left(\mathring{\Psi}^{(i)},\tilde{\Psi}^{(i)}\right)$ now result from the
power-series solutions $\mathring{\bm{H}}^{(i)}(x)$ for the Heun equation,
truncated as in (216). The pairs
$\left(\mathring{\Psi}^{(j+4)},\tilde{\psi}^{(j+4)}\right)$ ($j=1,2,3,4$) can
be generated from $\left(\mathring{\psi}^{(j)},\tilde{\Psi}^{(j)}\right)$ by
the substitutions $\bm{l}\mapsto-\bm{l}-1$ and $\bm{m}\mapsto-\bm{m}-1$. The
Arscott condition is satisfied if
$\mathring{\bm{\gamma}}_{n}^{(i)}=\mathring{{\gamma}}_{n}^{(i)}<0$, in
opposition to solutions of Section 5.1.

In the recurrence relations
$\mathring{\bm{\alpha}}_{n}\mathring{\bm{b}}_{n+1}+\mathring{\bm{\beta}}_{n}\mathring{\bm{b}}_{n}+\mathring{\bm{\gamma}}_{n}\mathring{\bm{b}}_{n-1}=0$
and
$\tilde{\bm{\alpha}}_{n}\tilde{\bm{b}}_{n+1}+\tilde{\bm{\beta}}_{n}\tilde{\bm{b}}_{n}+\tilde{\bm{\gamma}}_{n}\tilde{\bm{b}}_{n-1}=0$,
formally we have

$\displaystyle\tilde{\bm{\alpha}}_{n}^{(i)}=\mathring{\bm{\alpha}}_{n}^{(i)},\qquad\tilde{\bm{\beta}}_{n}^{(i)}=\mathring{\bm{\beta}}_{n}^{(i)},\qquad\tilde{\bm{\gamma}}_{n}^{(i)}=\mathring{\bm{\gamma}}_{n}^{(i)}.$
(280)

After writing the four types of expansions, we discuss the Arscott condition
(82) and comment on the Lamé equation.

####  5.2.1 First type: even solutions, period $2K$

$\displaystyle\begin{cases}\mathring{\bm{\Psi}}^{(1)}(u)=\mathrm{dn}^{\bm{l}+1}u\;\displaystyle\sum_{n=0}^{\frac{\bm{m-l}-1}{2}}\mathring{\bm{b}}_{n}^{(1)}\;\mathrm{cn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}-\bm{l}=1,3,5,\cdots,\end{array}\\\
\begin{array}[]{l}\hskip
199.16928pt\left[\;\bm{m}>-\frac{1}{2},\;\bm{m+l}>-2\;\right],\end{array}\vspace{2mm}\\\
\tilde{\bm{\Psi}}^{(1)}(u)=\mathrm{dn}^{\bm{l}+1}u\;\displaystyle\sum_{n=0}^{-\frac{\bm{m+l}+2}{2}}\tilde{\bm{b}}_{n}^{(1)}\;\mathrm{cn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}+\bm{l}=-2,-4,-6,\cdots,\end{array}\\\
\begin{array}[]{l}\hskip
199.16928pt\left[\;\bm{m}<-\frac{1}{2},\;\bm{m-l}<1\;\right],\end{array}\end{cases}$
(281a)
$\displaystyle\begin{array}[]{l}\mathring{\bm{\alpha}}_{n}^{(1)}=\left(1-\frac{1}{k^{2}}\right)\left(n+\frac{1}{2}\right)(n+1),\qquad\mathring{\bm{\beta}}_{n}^{(1)}=\left(\frac{1}{k^{2}}-2\right)n^{2}-(\bm{l}+1)n-\frac{{\cal
E}}{4k^{2}}+,\end{array}\vspace{2mm}$ (281c)
$\displaystyle\begin{array}[]{l}\frac{\bm{m}(\bm{m}+1)-\bm{l}-1}{4},\qquad\mathring{\bm{\gamma}}_{n}^{(1)}=\left(n-\frac{\bm{m}-\bm{l}+1}{2}\right)\left(n+\frac{\bm{l}+\bm{m}}{2}\right);\end{array}$
(281e)

As in the expansions in series of $\mathrm{sn}^{2}u$, there is only one
expansion when ($M_{1}$ and $N_{1}$ denote positive integers)

$\displaystyle\bm{m-l}-1=2M_{1}\;\mbox{ and
}\;\bm{m+l}+2=-2N_{1}\quad\left(\Leftrightarrow\bm{m}=M_{1}-N_{1}-\frac{1}{2},\right.:\vspace{2mm}$
$\displaystyle\left.\bm{l}=-M_{1}-N_{1}-\frac{3}{2}\right):\mathring{\bm{\psi}}^{(1)}\mbox{
if }\;M_{1}<N_{1},\qquad\tilde{\bm{\psi}}^{(1)}\mbox{ if }\;N_{1}<M_{1}.$

These expansions do not satisfy the Arscott condition. Actually, the Arscott
condition is not satisfied by series of $\mathrm{cn}^{2}u$ for any of the
cases given in (224g), (225h), $\cdots$, (231g). Thus, these cases will not be
considered from now on.

$\displaystyle\begin{cases}\mathring{\bm{\Psi}}^{(5)}(u)=\displaystyle\frac{1}{\mathrm{dn}^{\bm{l}}u}\;\sum_{n=0}^{\frac{\bm{l}-\bm{m}-1}{2}}\mathring{\bm{b}}_{n}^{(5)}\;\mathrm{cn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{l}-\bm{m}=1,3,5,\cdots\quad\hskip
14.22636pt;\end{array}\\\ \begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}<-\frac{1}{2},\;\bm{m+l}<0\;\right],\end{array}\vspace{2mm}\\\
\tilde{\bm{\Psi}}^{(5)}(u)=\displaystyle\frac{1}{\mathrm{dn}^{\bm{l}}u}\;\sum_{n=0}^{\frac{\bm{l}+\bm{m}}{2}}\;\tilde{\bm{b}}_{n}^{(5)}\;\mathrm{cn}^{2n}u\qquad\mbox{if
}\quad\begin{array}[]{l}\bm{l}+\bm{m}=0,2,4,\cdots,\qquad;\end{array}\\\
\begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}>-\frac{1}{2},\;\bm{l-m}<1\;\right],\end{array}\end{cases}$
(282a)
$\displaystyle\begin{array}[]{l}\mathring{\bm{\alpha}}_{n}^{(5)}=\left(1-\frac{1}{k^{2}}\right)\left(n+\frac{1}{2}\right)(n+1),\qquad\mathring{\bm{\beta}}_{n}^{(5)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\bm{l}n-\frac{{\cal
E}}{4k^{2}}+\end{array}\vspace{2mm}$ (282c)
$\displaystyle\begin{array}[]{l}\frac{\bm{l}+\bm{m}(\bm{m}+1)}{4},\qquad\mathring{\bm{\gamma}}_{n}^{(5)}=\left(n\textasciiacute-\frac{\bm{l}-\bm{m}+1}{2}\right)\left(n-\frac{\bm{l}+\bm{m}+2}{2}\right).\end{array}$
(282e)

####  5.2.2 Second type: odd solutions, period $4K$

$\displaystyle\begin{cases}\mathring{\bm{\Psi}}^{(2)}(u)=\mathrm{sn}u\;\mathrm{dn}^{\bm{l}+1}u\displaystyle\sum_{n=0}^{\frac{\bm{m-l}-2}{2}}\mathring{\bm{b}}_{n}^{(2)}\;\mathrm{cn}^{2n}u\qquad\mbox{
if }\quad\begin{array}[]{l}\bm{m}-\bm{l}=2,4,6,\cdots,\end{array}\\\
\begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}>-\frac{1}{2},\;\bm{m+l}>-3\;\right],\end{array}\vspace{2mm}\\\
{\tilde{\bm{\Psi}}}^{(2)}(u)=\mathrm{sn}u\;\mathrm{dn}^{\bm{l}+1}u\displaystyle\sum_{n=0}^{-\frac{\bm{m+l}+3}{2}}\tilde{\bm{b}}_{n}^{(2)}\;\mathrm{cn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}+\bm{l}=-3,-5,-7,\cdots,\end{array}\\\
\begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}<-\frac{1}{2},\;\bm{m-l}<2\;\right],\end{array}\end{cases}$
(283a)

$\displaystyle\begin{array}[]{l}\mathring{\bm{\alpha}}_{n}^{(2)}=\left(1-\frac{1}{k^{2}}\right)\left(n+\frac{1}{2}\right)(n+1),\qquad\mathring{\bm{\beta}}_{n}^{(2)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\left(\frac{1}{k^{2}}-\bm{l}-2\right)n+\end{array}\vspace{2mm}$
(283c) $\displaystyle\begin{array}[]{l}\frac{1-{\cal
E}}{4k^{2}}+\frac{\bm{m}(\bm{m}+1)-\bm{l}-2}{4},\qquad\mathring{\bm{\gamma}}_{n}^{(2)}=\left(n-\frac{\bm{m}-\bm{l}}{2}\right)\left(n+\frac{\bm{l}+\bm{m}+1}{2}\right);\end{array}$
(283e)
$\displaystyle\begin{cases}\mathring{\bm{\Psi}}^{(6)}(u)=\displaystyle\frac{\mathrm{sn}{u}}{{\rm
dn}^{\bm{l}}u}\;\sum_{n=0}^{\frac{\bm{l}-\bm{m}-2}{2}}\mathring{\bm{b}}_{n}^{(6)}\;{\rm
cn}^{2n}u\qquad\mbox{ if }\qquad\bm{l}-\bm{m}=2,4,6,\cdots,\\\
\begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}<-\frac{1}{2},\;\bm{m+l}<1\;\right],\end{array}\vspace{2mm}\\\
\tilde{\bm{\Psi}}^{(6)}(u)=\displaystyle\frac{\mathrm{sn}{u}}{{\rm
dn}^{\bm{l}}u}\;\sum_{n=0}^{\frac{\bm{l}+\bm{m}-1}{2}}\tilde{\bm{b}}_{n}^{(6)}\;{\rm
cn}^{2n}u\qquad\mbox{ if }\qquad\bm{l}+\bm{m}=1,3,5,\cdots,\\\
\begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}>-\frac{1}{2},\;\bm{l-m}<2\;\right],\end{array}\end{cases}$
(284a)
$\displaystyle\begin{array}[]{l}\mathring{\bm{\alpha}}_{n}^{(6)}=\left(1-\frac{1}{k^{2}}\right)\left(n+\frac{1}{2}\right)(n+1),\qquad\mathring{\bm{\beta}}_{n}^{(6)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\left(\frac{1}{k^{2}}+\bm{l}-1\right)n+\end{array}\vspace{3mm}$
(284c) $\displaystyle\begin{array}[]{l}\frac{1-{\cal
E}}{4k^{2}}+\frac{\bm{m}(\bm{m}+1)+\bm{l}-1}{4},\qquad\mathring{\bm{\gamma}}_{n}^{(6)}=\left(n-\frac{\bm{l}-\bm{m}}{2}\right)\left(n-\frac{\bm{l}+\bm{m}+1}{2}\right).\end{array}$
(284e)

####  5.2.3 Third type: even solutions, period $4K$

$\displaystyle\begin{cases}\mathring{\bm{\Psi}}^{(3)}(u)=\mathrm{cn}u\;\mathrm{dn}^{\bm{l}+1}u\;\displaystyle\sum_{n=0}^{\frac{\bm{m-l}-2}{2}}\mathring{\bm{b}}_{n}^{(3)}\;\mathrm{cn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}-\bm{l}=2,4,6,\cdots,\end{array}\\\
\begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}>-\frac{1}{2},\;\bm{m+l}>-3\;\right],\end{array}\vspace{2mm}\\\
{\tilde{\bm{\Psi}}}^{(3)}(u)=\mathrm{cn}u\;\mathrm{dn}^{\bm{l}+1}u\;\displaystyle\sum_{n=0}^{-\frac{\bm{m+l}+3}{2}}\tilde{\bm{b}}_{n}^{(3)}\;\mathrm{cn}^{2n}u\quad\mbox{
if }\quad\begin{array}[]{l}\bm{m}+\bm{l}=-3,-5,-7,\cdots,\end{array}\\\
\begin{array}[]{l}\hskip
227.62204pt\left[\;\bm{m}<-\frac{1}{2},\;\bm{m-l}<2\;\right],\end{array}\end{cases}$
(285a)
$\displaystyle\begin{array}[]{l}\mathring{\bm{\alpha}}_{n}^{(3)}=\left(1-\frac{1}{k^{2}}\right)\left(n+\frac{3}{2}\right)(n+1),\quad\mathring{\bm{\beta}}_{n}^{(3)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\left(\frac{1}{k^{2}}-\bm{l}-3\right)n+\end{array}\vspace{2mm}$
(285e) $\displaystyle\begin{array}[]{l}\frac{1-{\cal
E}}{4k^{2}}-\frac{1}{4}(3\bm{l}+5)+\frac{1}{4}\bm{m}(\bm{m}+1),\qquad\mathring{\bm{\gamma}}_{n}^{(3)}=\left(n-\frac{\bm{m}-\bm{l}}{2}\right)\left(n+\frac{\bm{l}+\bm{m}+1}{2}\right);\end{array}$
$\displaystyle\begin{cases}\mathring{\bm{\Psi}}^{(7)}(u)=\displaystyle\;\frac{\mathrm{cn}{u}}{{\rm
dn}^{\bm{l}}u}\;\sum_{n=0}^{\frac{\bm{l}-\bm{m}-2}{2}}\mathring{\bm{b}}_{n}^{(7)}\;{\rm
cn}{u}^{2n}u\quad\mbox{ if }\quad\bm{l}-\bm{m}=2,4,6,\cdots,\\\
\begin{array}[]{l}\hskip
199.16928pt\left[\;\bm{m}<-\frac{1}{2},\;\bm{m+l}<1\;\right],\end{array}\vspace{2mm}\\\
\tilde{\bm{\Psi}}^{(7)}(u)=\displaystyle\;\frac{\mathrm{cn}{u}}{{\rm
dn}^{\bm{l}}u}\;\sum_{n=0}^{\frac{\bm{l}+\bm{m}-1}{2}}\tilde{\bm{b}}_{n}^{(7)}\;{\rm
cn}{u}^{2n}u\quad\mbox{ if }\quad\bm{l}+\bm{m}=1,3,5,\cdots,\quad\\\
\begin{array}[]{l}\hskip
199.16928pt\left[\;\bm{m}>-\frac{1}{2},\;\bm{l-m}<2\;\right],\end{array}\end{cases}$
(286a)
$\displaystyle\begin{array}[]{l}\mathring{\bm{\alpha}}_{n}^{(7)}=\left(1-\frac{1}{k^{2}}\right)\left(n+\frac{3}{2}\right)(n+1),\qquad\mathring{\bm{\beta}}_{n}^{(7)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\left(\frac{1}{k^{2}}+\bm{l}-2\right)n+\end{array}\vspace{2mm}$
(286c) $\displaystyle\begin{array}[]{l}\frac{1-{\cal
E}}{4k^{2}}+\frac{1}{4}(3\bm{l}-2)+\frac{1}{4}\bm{m}(\bm{m}+1),\qquad\mathring{\bm{\gamma}}_{n}^{(7)}=\left(n-\frac{\bm{l}-\bm{m}}{2}\right)\left(n-\frac{\bm{l}+\bm{m}+1}{2}\right).\end{array}$
(286e)

####  5.2.4 Fourth type: odd solutions, period $2K$

$\displaystyle\begin{cases}\mathring{\bm{\Psi}}^{(4)}(u)=\mathrm{sn}u\;\mathrm{cn}u\;\mathrm{dn}^{\bm{l}+1}u\displaystyle\sum_{n=0}^{\frac{\bm{m-l}-3}{2}}\mathring{\bm{b}}_{n}^{(4)}\;\mathrm{cn}^{2n}u\quad\mbox{if}\quad\begin{array}[]{l}\bm{m}-\bm{l}=3,5,7,\cdots,\end{array}\\\
\begin{array}[]{l}\hskip
199.16928pt\left[\bm{m}>-\frac{1}{2},\;\bm{m+l}>-4\;\right],\end{array}\vspace{2mm}\\\
\tilde{\bm{\Psi}}^{(4)}(u)=\mathrm{sn}u\;\mathrm{cn}u\;\mathrm{dn}^{\bm{l}+1}u\displaystyle\sum_{n=0}^{-\frac{\bm{m+l}+4}{2}}\tilde{\bm{b}}_{n}^{(4)}\;\mathrm{cn}^{2n}u\;\mbox{
if }\;\begin{array}[]{l}\bm{m}+\bm{l}=-4,-6,-8,\cdots,\end{array}\\\
\begin{array}[]{l}\hskip
199.16928pt\left[\;\bm{m}<-\frac{1}{2},\;\bm{m-l}<3\;\;\right],\end{array}\end{cases}$
(287a)
$\displaystyle\begin{array}[]{l}\mathring{\bm{\alpha}}_{n}^{(4)}=\left(1-\frac{1}{k^{2}}\right)\left(n+\frac{3}{2}\right)(n+1),\quad\mathring{\bm{\beta}}_{n}^{(4)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\left(\frac{1}{k^{2}}-\bm{l}-4\right)n+\end{array}\vspace{3mm}$
(287c) $\displaystyle\begin{array}[]{l}\frac{4-{\cal
E}}{4k^{2}}-\frac{1}{4}(3\bm{l}+8)+\frac{1}{4}\bm{m}(\bm{m}+1),\quad\mathring{\bm{\gamma}}_{n}^{(4)}=\left(n-\frac{\bm{m}-\bm{l}-1}{2}\right)\left(n+\frac{\bm{l}+\bm{m}+2}{2}\right);\end{array}$
(287e)
$\displaystyle\begin{cases}\mathring{\bm{\Psi}}^{(8)}(u)\displaystyle\;=\frac{\mathrm{sn}{u}\;\mathrm{cn}{u}}{{\rm
dn}^{\bm{l}}u}\sum_{n=0}^{\frac{\bm{l}-\bm{m}-3}{2}}\mathring{\bm{b}}_{n}^{(8)}\;{\rm
cn}^{2n}u\quad\mbox{ if }\quad\bm{l}-\bm{m}=3,5,7,\cdots,\quad\\\
\begin{array}[]{l}\hskip
199.16928pt\left[\;\bm{m}<-\frac{1}{2},\;\bm{m+l}<2\;\right],\end{array}\vspace{2mm}\\\
\tilde{\bm{\Psi}}^{(8)}(u)=\displaystyle\;\frac{\mathrm{sn}{u}\;\mathrm{cn}{u}}{{\rm
dn}^{\bm{l}}u}\sum_{n=0}^{\frac{\bm{l}+\bm{m}-2}{2}}\tilde{\bm{b}}_{n}^{(8)}\;{\rm
cn}^{2n}u\quad\mbox{ if }\quad\bm{l}+\bm{m}=2,4,6,\cdots,\quad\\\
\begin{array}[]{l}\hskip
199.16928pt\left[\;\bm{m}>-\frac{1}{2},\;\bm{l-m}<3\;\;\right],\end{array}\end{cases}$
(288a)
$\displaystyle\begin{array}[]{l}\mathring{\bm{\alpha}}_{n}^{(8)}=\left(1-\frac{1}{k^{2}}\right)\left(n+\frac{3}{2}\right)(n+1),\qquad\mathring{\bm{\beta}}_{n}^{(8)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\left(\frac{2}{k^{2}}+\bm{l}-3\right)n+\end{array}\vspace{2mm}$
(288e) $\displaystyle\begin{array}[]{l}\frac{4-{\cal
E}}{4k^{2}}+\frac{3\bm{l}-5}{4}+\frac{\bm{m}}{4}(\bm{m}+1),\qquad\mathring{\bm{\gamma}}_{n}^{(8)}=\left(n-\frac{\bm{l}-\bm{m}-1}{2}\right)\left(n-\frac{\bm{l}+\bm{m}}{2}\right).\end{array}$

Concerning the Arscott condition (82), we notice that
$\left(1-{1}/{k^{2}}\right)<1$ implies that the coefficients
$\mathring{\bm{\alpha}}_{n}^{(i)}$ are negative. Then, the condition
$\mathring{\alpha}_{n-1}\mathring{\gamma}_{n}>0$ is satisfied if
$\mathring{\gamma}_{n}<0$ for any admissible value of $n$, that is,

$\displaystyle\begin{array}[]{r}\mathring{\bm{\gamma}}_{n}^{(1)}=\left(n-\frac{\bm{m}-\bm{l}+1}{2}\right)\left(n+\frac{\bm{l}+\bm{m}}{2}\right)<0\;\mbox{for}\;1\leq
n\leq\frac{\bm{m}-\bm{l}-1}{2},\;\left[\bm{m}-\bm{l}=3,5,7,\cdots\right],\end{array}$
$\displaystyle\begin{array}[]{r}\mathring{\bm{\gamma}}_{n}^{(2)}=\mathring{\bm{\gamma}}_{n}^{(3)}=\left(n-\frac{\bm{m}-\bm{l}}{2}\right)\left(n+\frac{\bm{l}+\bm{m}+1}{2}\right)<0\;\mbox{for}\;1\leq
n\leq\frac{\bm{m}-\bm{l}-2}{2},\;\left[\bm{m}-\bm{l}=4,6,8,\cdots\right],\end{array}$
$\displaystyle\begin{array}[]{r}\mathring{\bm{\gamma}}_{n}^{(4)}=\left(n-\frac{\bm{m}-\bm{l}-1}{2}\right)\left(n+\frac{\bm{l}+\bm{m}+2}{2}\right)<0\;\mbox{for}\qquad
1\leq
n\leq\frac{\bm{m}-\bm{l}-3}{2},\;\left[\bm{m}-\bm{l}=5,7,9,\cdots\right],\end{array}$

for ${\mathring{\Psi}}^{(i)}$. As the first term of these
$\mathring{\gamma}_{n}^{(i)}$ is negative, $\mathring{\gamma}_{n}^{(i)}<0$ if
the the last term is positive for each value of $n$. From (237) and similar
relations for each $\mathring{\gamma}_{n}^{(i)}$, it follows that
$\bm{m}>-{1}/{2}$. In addition, from $\mathring{\gamma}_{n=1}^{(i)}<0$ we get
restritions on $\bm{m}+\bm{l}$, namely,

$\displaystyle\begin{array}[]{l}\bm{m}>-\frac{1}{2}\mbox{ for
}\mathring{\bm{\Psi}}^{(i)},\quad\bm{m}+\bm{l}+2>0\;\mbox{ for
}\;\mathring{\bm{\Psi}}^{(1)},\;\;\bm{m}+\bm{l}+3>0\;\mbox{ for
}\;\mathring{\bm{\Psi}}^{(2)}\mbox{ and
}\mathring{\bm{\Psi}}^{(3)},\end{array}\vspace{3mm}$ (290)
$\displaystyle\begin{array}[]{l}\bm{m}+\bm{l}+4>0\;\mbox{ for
}\;\mathring{\bm{\Psi}}^{(4)}.\end{array}$ (292)

We see, for example, that the interval

$\displaystyle\begin{array}[]{l}-\frac{\bm{m}+\bm{l}}{2}<n<\frac{\bm{m}-\bm{l}-1}{2}+1,\quad\mbox{for
}\;\bm{m}>-\frac{1}{2}\;\;\mbox{ and }-\bm{m}-\bm{l}<2\end{array}$

assures that $\mathring{\bm{\gamma}}_{n}^{(1)}<0$ for all the possible values
for $n$.

On the other hand, for ${\tilde{\Psi}}^{(i)}$ we must require that
($\mathring{\gamma}_{n}^{(2)}=\mathring{\gamma}_{n}^{(3)}$)

$\displaystyle\begin{array}[]{r}{\mathring{\gamma}}_{n}^{(1)}=\left(n+\frac{\bm{l}+\bm{m}}{2}\right)\left(n+\frac{\bm{l}-\bm{m}-1}{2}\right)<0\;\mbox{for}\;1\leq
n\leq-\frac{\bm{m}+\bm{l}+2}{2},\;\left[\bm{m}+\bm{l}=-4,-6,-8,\cdots\right],\end{array}$
$\displaystyle\begin{array}[]{r}{\mathring{\gamma}}_{n}^{(2)}=\left(n+\frac{\bm{l}+\bm{m}+1}{2}\right)\left(n+\frac{\bm{l}-\bm{m}}{2}\right)<0\;\mbox{for}\;1\leq
n\leq-\frac{\bm{m}+\bm{l}+3}{2},\;\left[\bm{m}+\bm{l}=-5,-7,-9,\cdots\right],\end{array}$
$\displaystyle\begin{array}[]{r}{\mathring{\gamma}}_{n}^{(4)}=\left(n+\frac{\bm{l}+\bm{m}+2}{2}\right)\left(n+\frac{\bm{l}-\bm{m}+1}{2}\right)<0\;\mbox{for}\;1\leq
n\leq-\frac{\bm{m}+\bm{l}+4}{2},\;\left[\bm{m}+\bm{l}=-6.-8,\cdots\right].\end{array}$

Since the first term is negative, $\tilde{\gamma}_{n}^{(i)}<0$ if the last
term on the right of these equations is positive. Thence, we find that the
Arscott condition is assured by the following restrictions on the parameters

$\displaystyle\begin{array}[]{l}\bm{m}<-\frac{1}{2}\mbox{ for
}{\tilde{\Psi}}^{(i)},\;\;\bm{l}-\bm{m}+1>0\;\mbox{ for
}\;{\tilde{\Psi}}^{(1)},\;\;\bm{l}-\bm{m}+2>0\;\mbox{ for
}\;{\tilde{\Psi}}^{(2)}\mbox{ and
}{\tilde{\Psi}}^{(3)},\end{array}\vspace{3mm}$
$\displaystyle\begin{array}[]{l}\bm{l}-\bm{m}+3>0\;\mbox{ for
}\;{\tilde{\Psi}}^{(4)};\end{array}$ (296)

These conditions imply that $\mathring{\gamma}_{n}^{(i)}<0$ for any admissible
$n$. For example, by (237) and (5.2.4), we see that
$\mathring{\gamma}_{n}^{(1)}<0$ in the interval

$\displaystyle\begin{array}[]{l}\frac{\bm{m}-\bm{l}+1}{2}<n<-\frac{\bm{m}+\bm{l}}{2}\quad\mbox{for}\quad\bm{l}-\bm{m}+1>0\quad\mbox{and}\quad\bm{m}<-\frac{1}{2}\end{array}$

which includes all the possible values of $n$ for the solutions
${\tilde{\Psi}}^{(1)}$.

The eigenfunctions
$\left(\mathring{\bm{\Psi}}^{(i)},\tilde{\bm{\Psi}}^{(i)}\right)$ do not
satisfy the Arscott condition for the examples discussed in Section 5.1.
However, we can check that the sixteen eigenfunctions give different types of
solutions for Lamé equation (213) when $\bm{m}=0,\pm 1,\pm 2,\cdots$; each
solution satisfies the Arscott condition. Except if $\bm{m}=-2$, $\bm{m}=-1$,
$\bm{m}=0$ and $\bm{m}=1$, for each value of $\bm{m}$ there are four types of
solutions.

$\displaystyle\begin{array}[]{l}\mbox{even solutions, period
}2K\;(\bm{l}=0):\left\\{\begin{array}[]{l}\mathring{\bm{\Psi}}^{(1)}\quad\mbox{if}\quad\bm{m}=1,3,5,\cdots;\vspace{2mm}\\\
\tilde{\bm{\Psi}}^{(1)}\quad\mbox{if}\quad\bm{m}=-2,-4,-6,\cdots;\vspace{2mm}\\\
\mathring{\bm{\Psi}}^{(5)}\quad\mbox{if}\quad\bm{m}=-1,-3,-5,\cdots;\vspace{2mm}\\\
\tilde{\bm{\Psi}}^{(5)}\quad\mbox{if}\quad\bm{m}=0,2,4,\cdots;\end{array}\right.\vspace{2mm}\end{array}$
(303) $\displaystyle\begin{array}[]{l}\mbox{odd solutions, period
}2K\;(\bm{l}=0):\left\\{\begin{array}[]{l}\mathring{\bm{\Psi}}^{(4)}\quad\mbox{if}\quad\bm{m}=3,5,7,\cdots;\vspace{2mm}\\\
\tilde{\bm{\Psi}}^{(4)}\quad\mbox{if}\quad\bm{m}=-4,-6,-8,\cdots;\vspace{2mm}\\\
\mathring{\bm{\Psi}}^{(8)}\quad\mbox{if}\quad\bm{m}=-3,-5,-7,\cdots;\vspace{2mm}\\\
\tilde{\bm{\Psi}}^{(8)}\quad\mbox{if}\quad\bm{m}=2,4,6,\cdots;\end{array}\right.\vspace{2mm}\end{array}$
(309) $\displaystyle\begin{array}[]{l}\mbox{odd solutions, period
}4K\;(\bm{l}=0):\left\\{\begin{array}[]{l}\mathring{\bm{\Psi}}^{(2)}\;\mbox{
if }\;\bm{m}=2,4,6,\cdots;\vspace{2mm}\\\ \tilde{\bm{\Psi}}^{(2)}\;\mbox{ if
}\;\bm{m}=-3,-5,-7,\cdots;\vspace{2mm}\\\ \mathring{\bm{\Psi}}^{(6)}\;\mbox{
if }\;\bm{m}=-2,-4,-6,\cdots;\vspace{2mm}\\\ \tilde{\bm{\Psi}}^{(6)}\;\mbox{
if }\;\bm{m}=1,3,5,\cdots;\end{array}\right.\end{array}\vspace{2mm}$ (315)
$\displaystyle\begin{array}[]{l}\mbox{even solutions, period
}4K\;(\bm{l}=0):\left\\{\begin{array}[]{l}\tilde{\bm{\Psi}}^{(3)}\quad\mbox{if}\quad\bm{m}=2,4,6,\cdots;\vspace{2mm}\\\
\mathring{\bm{\Psi}}^{(3)}\quad\mbox{if}\quad\bm{m}=-3,-5,-7,\cdots;\vspace{2mm}\\\
\mathring{\bm{\Psi}}^{(7)}\quad\mbox{if}\quad\bm{m}=-2,-4,-6,\cdots;\vspace{2mm}\\\
\tilde{\bm{\Psi}}^{(7)}\quad\mbox{if}\quad\bm{m}=1,3,5,\cdots.\end{array}\right.\end{array}$
(321)

We find the same results if we take $\bm{l}=0$ or $\bm{l}=-1$. By taking
$\bm{l}=-1$ we simply rearrange the previous expansions. We obtain, for
instance,

$\mathring{\bm{\Psi}}^{(1)}\Big{|}_{\bm{l}=-1}=\tilde{\bm{\Psi}}^{(5)}\Big{|}_{\bm{l}=0},\;\tilde{\bm{\Psi}}^{(1)}\Big{|}_{\bm{l}=-1}=\mathring{\bm{\Psi}}^{(5)}\Big{|}_{\bm{l}=0},\;\mathring{\bm{\Psi}}^{(5)}\Big{|}_{\bm{l}=-1}=\tilde{\bm{\Psi}}^{(1)}\Big{|}_{\bm{l}=0},\;\tilde{\bm{\Psi}}^{(5)}\Big{|}_{\bm{l}=-1}=\tilde{\bm{\Psi}}^{(1)}\Big{|}_{\bm{l}=0}.$

Other examples, for fixed values of $\bm{m}$ and $\bm{l}$, are presented in
Appendix C.

###  5.3 Finite series of hypergeometric functions ${\psi}^{(i)}(u)$ when
$\bm{l}$ is an integer and $\bm{m}\neq-1/2$ is half an odd integer

If $\bm{m}$ is half an odd positive integer, finite-series solutions
${\psi}^{(i)}(u)$ with $0\leq n\leq\bm{m}-1/2$ are obtained by inserting in
(205) the hypergeometric-function expansions ${H}^{(i)}(x)$ with $i=1,4,6,7$.
In these cases, the series are finite because the factor
$(n+\alpha-\beta)=(n-\bm{m}-1/2)$ implies $\gamma_{n}^{(i)}=0$ for
$n=\bm{m}+1/2$ as demanded in (75). Since $\delta=1/2$, according to (178) the
four hypergeometric functions converge when $z=1$ or $\mathrm{sn}^{2}u=1$,
where $z={(a-1)x}/{(a-x)}=(1-k^{2})\;\mathrm{sd}^{2}u$.

If $\bm{m}\neq-1/2$ is half an odd negative integer, finite-series solutions
${\psi}^{(i)}$ with $0\leq n\leq-\bm{m}-3/2$ result from ${H}^{(i)}$ for
$i=2,3,5,8$. Now the series terminate at $n=-\bm{m}-3/2$ because