$(n-\alpha+\beta)=(n+\bm{m}+1/2)$ implies $\gamma_{n}^{(i)}=0$ for
$n=-\bm{m}-1/2$ as demanded in (75). Since $\delta=1/2$, by (178) the four
solutions converge when $z=1$ ($\mathrm{sn}^{2}u=1$).

Eigenfunctions for negative and positive values of $\bm{m}$ are connected to
each other by means of the substitutions (206). In both cases (positive and
negative $\bm{m}$) the solutions are degenerate in the sense that there are
two independent solutions with the same energies ${\cal E}$ for a given set of
values for the parameters ($\bm{m}$, $\bm{l}$, $k^{2}$). In addition, for a
given expansion the the Arscott condition (82) is satisfied only if we impose
restrictions on the values of $\bm{l}$ and $\bm{m}$. However, by considering
all the expansions, we find at least one expansion satisfying the condition
for any allowed value of ($\bm{m}$, $\bm{l}$).

Here and in Sec. 5.4, in order to check the parity and periodicity properties
we consider the one-term expansions because in these cases the hypergeometric
functions can be written in terms of elementary functions rather than in terms
of infinite series. For this we use [3]

$\displaystyle\begin{array}[]{l}F\left(\mathrm{a},-\mathrm{a};\frac{1}{2};-z^{2}\right)=\frac{1}{2}\left[\left(\sqrt{1+z^{2}}+z\right)^{2\mathrm{a}}+\left(\sqrt{1+z^{2}}-z\right)^{2\mathrm{a}}\right],\vspace{3mm}\\\
F\left(\mathrm{a},1-\mathrm{a};\frac{1}{2};-z^{2}\right)=\frac{1}{2\sqrt{1+z^{2}}}\left[\left(\sqrt{1+z^{2}}+z\right)^{2\mathrm{a}-1}+\left(\sqrt{1+z^{2}}-z\right)^{2\mathrm{a}-1}\right],\vspace{3mm}\\\
F\left(\mathrm{a},1-\mathrm{a};\frac{3}{2};-z^{2}\right)=\frac{1}{(2-4\mathrm{a})z}\left[\left(\sqrt{1+z^{2}}+z\right)^{1-2\mathrm{a}}-\left(\sqrt{1+z^{2}}-z\right)^{1-2\mathrm{a}}\right].\end{array}$
(325)

These relations can also be applied to some two-term series. Further, we find
that such parity and periodicity properties hold for any finite-series
expansion.

####  5.3.1 Finite series when $\bm{l}$ is an integer and $\bm{m}$ is half an
odd positive integer

First we write the eigenfunctions ${\psi}^{(1)}(u)$ and ${\psi}^{(6)}(u)$.
Then, we proof the degeneracy by using the matrix form for the recurrence
relations. After this, we discuss the Arscott condition and, finally, we
consider the eigenfunctions ${\psi}^{(4)}(u)$ and ${\psi}^{(7)}(u)$.

$\displaystyle\begin{cases}{\psi}^{(1)}(u)\stackrel{{\scriptstyle\text{(\ref{adendos-1},\ref{second-
ganguly-3})}}}{{=}}\left(\mathrm{dn}\,u\right)^{\bm{m}}\displaystyle\sum_{n=0}^{\bm{m}-1/2}\begin{array}[]{l}{b}_{n}^{(1)}\left[(1-k^{2})\;\mathrm{sd}^{2}u\right]^{n}\times\end{array}\\\
\begin{array}[]{l}\tilde{F}\left[n+\frac{\bm{l}-\bm{m}+1}{2},\frac{\bm{m}-\bm{l}-1}{2};n+\frac{1}{2};(1-k^{2})\;\mathrm{sd}^{2}u\right],\qquad\bm{m}=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots,\quad[\mbox{even,\;
{4K}}],\end{array}\vspace{3mm}\\\
{\alpha}_{n}^{(1)}=\left(1-\frac{1}{k^{2}}\right)(n+1),\;{\beta}_{n}^{(1)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\left[\left(\frac{1}{k^{2}}-1\right)(\bm{l-m}+1)+\bm{m}\right]n+\vspace{3mm}\\\
\frac{(\bm{l}+1-\bm{m})^{2}-{\cal
E}}{4k^{2}}+\frac{2\bm{lm}-\bm{l}+\bm{m}-(\bm{m}-1)^{2}}{4},\quad{\gamma}_{n}^{(1)}=\left(n+\frac{\bm{l-m-1}}{2}\right)\left(n+\frac{\bm{l-m}}{2}\right)\left(n-\bm{m}-\frac{1}{2}\right);\end{cases}$
(326)
$\displaystyle\begin{cases}{\psi}^{(6)}(u)\stackrel{{\scriptstyle\text{(\ref{adendos-6},\ref{second-
ganguly-3})}}}{{=}}\mathrm{sn}u(\mathrm{dn}u)^{m-1}\displaystyle\sum_{n=0}^{\bm{m}-{1}/{2}}\begin{array}[]{l}{b}_{n}^{(6)}\left[(1-k^{2})\;\mathrm{sd}^{2}u\right]^{n}\times\end{array}\vspace{2mm}\\\
\tilde{F}\left[n-\frac{\bm{l}+\bm{m}-1}{2},\frac{\bm{m}+\bm{l}+1}{2};n+\frac{3}{2};(1-k^{2})\;\mathrm{sd}^{2}u\right],\quad\bm{m}=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots,\;[\mbox{odd,\;{
4K}}],\vspace{3mm}\\\
{\alpha}_{n}^{(6)}=\left(1-\frac{1}{k^{2}}\right)(n+1),\quad{\beta}_{n}^{(6)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\left[\left(1-\frac{1}{k^{2}}\right)(\bm{l+m})+\bm{m}-1\right]n+\vspace{3mm}\\\
\frac{(\bm{l}+\bm{m})^{2}-{\cal
E}}{4k^{2}}+\frac{\bm{l}-2\bm{lm}+\bm{m}-(\bm{m}-1)^{2}}{4},\qquad{\gamma}_{n}^{(6)}=\left(n-\frac{\bm{l+m+1}}{2}\right)\left(n-\frac{\bm{l+m}}{2}\right)\left(n-\bm{m}-\frac{1}{2}\right).\end{cases}$
(327)

For $\bm{m}=1/2$ we obtain two independent eigenfunctions,

$\displaystyle\begin{array}[]{l}{\psi}^{(1)}(u)\Big{|}_{\bm{m}=1/2}={b}_{0}^{(1)}\sqrt{\mathrm{dn}\,u}\;\tilde{F}\left[\frac{1}{2}\left(\bm{l}+\frac{1}{2}\right),-\frac{1}{2}\left(\bm{l}+\frac{1}{2}\right);\frac{1}{2};(1-k^{2})\;\mathrm{sd}^{2}u\right],\vspace{3mm}\\\
{\psi}^{(6)}(u)\Big{|}_{\bm{m}=1/2}={b}_{0}^{(6)}\frac{\mathrm{sn}u}{\sqrt{\mathrm{dn}u}}\;\tilde{F}\left[-\frac{1}{2}\left(\bm{l}-\frac{1}{2}\right),\frac{1}{2}\left(\bm{l}+\frac{3}{2}\right);\frac{3}{2};(1-k^{2})\;\mathrm{sd}^{2}u\right],\end{array}$

which are degenerate because both correspond to the same energy ${\cal E}$,
that is,

$\displaystyle\begin{array}[]{l}{\beta}_{0}^{(1)}=0,\;\quad{\beta}_{0}^{(6)}=0\quad\Rightarrow\quad{\cal
E}=\left(\bm{l}+\frac{1}{2}\right)^{2}+\frac{k^{2}}{4},\quad\left[\bm{m}=\frac{1}{2}\right].\end{array}$

To prove the degeneracy for $\bm{m}\geq 3/2$ , first we redefine
${b}_{n}^{(6)}$ as

$\displaystyle\begin{array}[]{l}{b}_{n}^{(6)}=\left(\frac{k^{2}}{1-k^{2}}\right)^{n}\Gamma\left(n-\frac{\bm{l+m}-1}{2}\right)\Gamma\left(n-\frac{\bm{l+m}-2}{2}\right)\hat{b}_{n},\end{array}$
(330b)

which implies that the $\hat{b}_{n}$ obey the relations
$\hat{\alpha}_{n}\hat{b}_{n+1}+\hat{\beta}_{n}\hat{b}_{n}+\hat{\gamma}_{n}\hat{b}_{n-1}=0$
($\hat{b}_{-1}=0$) where

$\displaystyle\begin{array}[]{l}\hat{{\alpha}}_{n}=-\left(n-\frac{\bm{l+m-1}}{2}\right)\left(n-\frac{\bm{l+m}-2}{2}\right)(n+1),\qquad\hat{\beta}_{n}={\beta}_{n}^{(6)},\end{array}\vspace{2mm}$
(330d)
$\displaystyle\begin{array}[]{l}\hat{{\gamma}}_{n}=-\left(1-\frac{1}{k^{2}}\right)\left(n-\bm{m}-\frac{1}{2}\right).\end{array}$
(330f)

We show the degeneracy by writing the relations for $b_{n}^{(1)}$ and
$\hat{b}_{n}$ in the forms $\mathbb{A}\vec{b}=0$ and
$\mathbb{B}\vec{\hat{b}}=0$, where

$\displaystyle\begin{array}[]{l}\vec{b}=\left(b_{0}^{(1)},b_{1}^{(1)},\cdots,b_{\bm{m}-\frac{1}{2}}^{(1)}\right)^{t},\qquad\vec{\hat{b}}=\left(\hat{b}_{0},\hat{b}_{1},\cdots,\hat{b}_{\bm{m}-\frac{1}{2}}\right)^{t},\end{array}$

while $\mathbb{A}$ and $\mathbb{B}$ denote the matrices

$\displaystyle\mathbb{A}=\left[\begin{array}[]{ccccccccc}{\beta}_{0}^{(1)}&{\alpha}_{0}^{(1)}&0&&\vspace{2mm}\vspace{2mm}\hfil\\\
{\gamma}_{1}^{(1)}&{\beta}_{1}^{(1)}&{\alpha}_{1}^{(1)}&\vspace{2mm}\hfil\\\
&\ddots&\ddots&\ddots&\vspace{2mm}\hfil\\\
&&{\gamma}_{\bm{m}-\frac{3}{2}}^{(1)}&{\beta}_{\bm{m}-\frac{3}{2}}^{(1)}&{\alpha}_{\bm{m}-\frac{3}{2}}^{(1)}\vspace{2mm}\\\
&&0&{\gamma}_{\bm{m}-\frac{1}{2}}^{(1)}&{\beta}_{\bm{m}-\frac{1}{2}}^{(1)}\end{array}\right],\;{\mathbb{B}}=\left[\begin{array}[]{ccccccccc}\hat{\beta}_{0}&\hat{\alpha}_{0}&0&&\vspace{2mm}\vspace{2mm}\hfil\\\
\hat{\gamma}_{1}&\hat{\beta}_{1}&\hat{\alpha}_{1}&\vspace{2mm}\hfil\\\
&\ddots&\ddots&\ddots&\vspace{2mm}\hfil\\\
&&\hat{\gamma}_{\bm{m}-\frac{3}{2}}&\hat{\beta}_{\bm{m}-\frac{3}{2}}&\hat{\alpha}_{\bm{m}-\frac{3}{2}}\vspace{2mm}\\\
&&0&\hat{\gamma}_{\bm{m}-\frac{1}{2}}&\hat{\beta}_{\bm{m}-\frac{1}{2}}\end{array}\right].$

Then (326), (327) and (330d) imply that both matrices have the same elements
since

$\displaystyle{\mathbb{B}}=\left[\begin{array}[]{ccccccccc}\beta_{\bm{m}-\frac{1}{2}}^{(1)}&\gamma_{\bm{m}-\frac{1}{2}}^{(1)}&0&&\vspace{2mm}\vspace{2mm}\hfil\\\
\alpha_{\bm{m}-\frac{3}{2}}^{(1)}&\beta_{\bm{m}-\frac{3}{2}}^{(1)}&\gamma_{\bm{m}-\frac{3}{2}}^{(1)}&\vspace{2mm}\hfil\\\
&\ddots&\ddots&\ddots&\vspace{2mm}\hfil\\\
&&\alpha_{1}^{(1)}&\beta_{1}^{(1)}&\gamma_{1}^{(1)}\vspace{2mm}\\\
&&0&\alpha_{0}^{(1)}&\beta_{0}^{(1)}\end{array}\right].$ (336)

The degeneracy results from the fact that $\det{\mathbb{A}}=$
$\det{\mathbb{B}}$. In effect, let $\mathbb{U}$ be a square matrix of order
$\left(\bm{m}+\frac{1}{2}\right)$, having $1$’s on the antidiagonal as the
only nonzero elements, that is,

$\displaystyle\mathbb{U}=\mathbb{U}^{-1}=\left[\begin{array}[]{ccc}&&1\\\
&.&\\\ 1&&\end{array}\right],\qquad\det{\mathbb{U}}=\pm 1,$ (340)

where $\mathbb{U}^{-1}$ is the matrix inverse to $\mathbb{U}$. Then, we find
that

$\displaystyle{\mathbb{B}}=\mathbb{U}\mathbb{A}\mathbb{U}\qquad\Rightarrow\qquad\det{\mathbb{A}}=\det{\mathbb{B}},$
(341)

where the last equality follows from the properties of the determinants. The
above approach has already been used to get the degeneracy of wavefunctions
for a problem which is ruled by a reduced confluent Heun equation [39].

Finally, in (326) and (327) ${{\alpha}}_{n}^{(1)}={{\alpha}}_{n}^{(6)}<0$.
Thence, the Arscott condition (82) for $\bm{m}\geq 3/2$ requires that
${{\gamma}}_{n}^{(1)}<0$ and ${{\gamma}}_{n}^{(6)}<0$. From the expressions
for ${{\gamma}}_{n}^{(1)}$ and ${{\gamma}}_{n}^{(6)}$ these conditions are
equivalent to

$\displaystyle\begin{array}[]{l}f(n)=\left(n+\frac{\bm{l-m}-1}{2}\right)\left(n+\frac{\bm{l-m}}{2}\right)>0\quad\mbox{for}\quad{\psi}^{(1)},\end{array}$
(343)
$\displaystyle\begin{array}[]{l}g(n)=\left(n-\frac{\bm{l+m}+1}{2}\right)\left(n-\frac{\bm{l+m}}{2}\right)>0\quad\mbox{for}\quad{\psi}^{(6)}\end{array}$
(345)

for any admissible value of $n$, that is, $1\leq n\leq{\bm{m}}-\frac{1}{2}$.
However, these conditions are satisfied only for special values of $\bm{l}$
and $\bm{m}$. As an illustration, for the Lamé equation ($\bm{l}=0$,
$\bm{l}=-1$), we find that

$\displaystyle\begin{array}[]{l}\mbox{for}\;\bm{l}=0,\quad
f(n)=g(n)>0\quad\mbox{ only if
}\quad\bm{m}=\frac{5}{2},\frac{9}{2},\frac{13}{2},\cdots;\vspace{3mm}\\\
\mbox{for}\;\bm{l}=-1\;(f\neq g),\quad f(n)>0\mbox{ and }g(n)>0\quad\mbox{
only if }\quad\bm{m}=\frac{3}{2},\frac{7}{2},\frac{11}{2},\cdots.\end{array}$
(348)

So, we get two degenerate finite-series solutions for the Lamé equation when
$\bm{m}$ is half an odd positive integer – for $\bm{m}=1/2$ see solutions
after Eq. (327). These are not the finite series proposed by Ince [40].

On the other side,

$\displaystyle\begin{array}[]{l}f(n)\leq 0\;\mbox{ if
}\;\frac{\bm{m-l}}{2}\leq n\leq\frac{\bm{m-l}}{2}+\frac{1}{2},\quad\mbox{and
}\;g(n)\leq 0\;\mbox{ if }\;\frac{\bm{m+l}}{2}\leq
n\leq\frac{\bm{m+l}}{2}+\frac{1}{2}.\end{array}$ (350)

Then, Arscott condition holds if the above intervals are empty (no admissible
$n$). In particular we can satisfy the condition by demanding that, for a
given value of $\bm{m}$, the upper bounds of the intervals (350) are less than
1, that is, $\bm{m-l}<1$ and $\bm{m+l}<1$. Since $\bm{m}=3/2,5/2.\cdots$, the
Arscott condition stands if

$\displaystyle\begin{array}[]{l}\bm{l>m-1}\mbox{ for
}{\psi}^{(1)}\;(\bm{l}\mbox{ is positive integer});\;\bm{l<1-m}\mbox{ for
}{\psi}^{(6)}\;(\bm{l}\mbox{ negative integer}).\end{array}$ (352)

In these cases, the degeneracy proof fails because $\bm{l}$ takes different
values in each expansion.

On the other hand, the degenerate eigenfunctions ${\psi}^{(4)}(u)$ and
${\psi}^{(7)}(u)$ are given by

$\displaystyle\begin{cases}{\psi}^{(4)}(u)\stackrel{{\scriptstyle\text{(\ref{adendos-4},\ref{second-
ganguly-3})}}}{{=}}\operatorname{sn}{u}\left(\mathrm{dn}\,u\right)^{\bm{m-1}}\displaystyle\sum_{n=0}^{\bm{m}-1/2}\begin{array}[]{l}{b}_{n}^{(4)}\left[(1-k^{2})\;\mathrm{sd}^{2}u\right]^{n}\times\end{array}\\\
\begin{array}[]{l}\tilde{F}\left[n+\frac{\bm{l}-\bm{m}+2}{2},\frac{\bm{m}-\bm{l}}{2};n+\frac{3}{2};(1-k^{2})\;\mathrm{sd}^{2}u\right],\quad\bm{m}=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots,\quad[\mbox{odd,\;
{4K}}],\end{array}\vspace{3mm}\\\
{\alpha}_{n}^{(4)}=\left(1-\frac{1}{k^{2}}\right)(n+1),\;{\beta}_{n}^{(4)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\left[\left(\frac{1}{k^{2}}-1\right)(\bm{l-m}+1)+\bm{m}-1\right]n+\vspace{3mm}\\\
\frac{(\bm{l}+1-\bm{m})^{2}-{\cal
E}}{4k^{2}}+\frac{2\bm{l}\bm{m}-\bm{m}^{2}+5\bm{m}-2-\bm{l}}{4},\quad{\gamma}_{n}^{(4)}=\left(n+\frac{\bm{l-m+1}}{2}\right)\left(n+\frac{\bm{l-m}}{2}\right)\left(n-\bm{m}-\frac{1}{2}\right);\end{cases}$
(353)
$\displaystyle\begin{cases}{\psi}^{(7)}(u)\stackrel{{\scriptstyle\text{(\ref{adendos-7},\ref{second-
ganguly-3})}}}{{=}}(\operatorname{dn}u)^{m}\displaystyle\sum_{n=0}^{\bm{m}-{1}/{2}}\begin{array}[]{l}{b}_{n}^{(7)}\left[(1-k^{2})\;\mathrm{sd}^{2}u\right]^{n}\times\end{array}\vspace{2mm}\\\
\tilde{F}\left[n-\frac{\bm{l}+\bm{m}}{2},\frac{\bm{m}+\bm{l}}{2};n+\frac{1}{2};(1-k^{2})\;\mathrm{sd}^{2}u\right],\quad\bm{m}=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots,\;[\mbox{even,\;{
4K}}],\vspace{3mm}\\\
{\alpha}_{n}^{(7)}=\left(1-\frac{1}{k^{2}}\right)(n+1),\quad{\beta}_{n}^{(7)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\left[\left(1-\frac{1}{k^{2}}\right)(\bm{l+m})+\bm{m}\right]n+\vspace{3mm}\\\
\frac{(\bm{l}+\bm{m})^{2}-{\cal
E}}{4k^{2}}+\frac{\bm{l}+\bm{m}-2\bm{lm}-\bm{m}^{2}}{4},\qquad{\gamma}_{n}^{(7)}=\left(n-\frac{\bm{l+m+2}}{2}\right)\left(n-\frac{\bm{l+m+1}}{2}\right)\left(n-\bm{m}-\frac{1}{2}\right).\end{cases}$
(354)

For one-term series ($\bm{m}=1/2$), ${\psi}^{(4)}$ and ${\psi}^{(7)}$ are not
independent respect to ${\psi}^{(1)}$ and ${\psi}^{(6)}$, since

$\displaystyle\begin{array}[]{l}{\psi}^{(4)}(u)={\psi}^{(6)}(u)={{b}_{0}^{(4)}}\sqrt{\operatorname{dn}{u}}\left[\left(\frac{\operatorname{cn}{u}+\sqrt{k^{2}-1}\operatorname{sn}{u}}{\operatorname{dn}{u}}\right)^{\bm{l}+\frac{1}{2}}-\left(\frac{\operatorname{cn}{u}-\sqrt{k^{2}-1}\operatorname{sn}{u}}{\operatorname{dn}{u}}\right)^{\bm{l}+\frac{1}{2}}\right],\qquad\bm{m}=\frac{1}{2},\vspace{3mm}\\\
{\psi}^{(7)}(u)={\psi}^{(1)}(u)={{b}_{0}^{(7)}\sqrt{\operatorname{dn}{u}}}\left[\left(\frac{\operatorname{cn}{u}+\sqrt{k^{2}-1}\operatorname{sn}{u}}{\operatorname{dn}{u}}\right)^{\bm{l}+\frac{1}{2}}+\left(\frac{\operatorname{cn}{u}-\sqrt{k^{2}-1}\operatorname{sn}{u}}{\operatorname{dn}{u}}\right)^{\bm{l}+\frac{1}{2}}\right],\qquad\bm{m}=\frac{1}{2},\end{array}$

which are odd and even solutions, respectively, both having period $4K$. The
previous expressions follow from relations (325).

To prove the degeneracy for $\bm{m}\geq 3/2$ , first we redefine
${b}_{n}^{(7)}$ as

$\displaystyle\begin{array}[]{l}{b}_{n}^{(7)}=\left(\frac{k^{2}}{1-k^{2}}\right)^{n}\Gamma\left(n-\frac{\bm{l+m}-1}{2}\right)\Gamma\left(n-\frac{\bm{l+m}}{2}\right)\tilde{b}_{n}^{(7)}\end{array}$

and obtain the relations

$\displaystyle\begin{array}[]{l}-\left(n-\frac{\bm{l+m}-1}{2}\right)\left(n-\frac{\bm{l+m}}{2}\right)(n+1)\;\tilde{b}_{n+1}^{(7)}+{\beta}_{n}^{(7)}\;\tilde{b}_{n}^{(7)}+\frac{1-k^{2}}{k^{2}}\left(n-\bm{m}-\frac{1}{2}\right)\tilde{b}_{n-1}^{(7)}=0.\end{array}$

Thence, if $\mathbb{C}$ and $\mathbb{D}$ represent the matrices corresponding
to the recurrence relations for ${b}_{n}^{(4)}$ and $\tilde{b}_{n}^{(7)}$, we
find that ${\mathbb{D}}=\mathbb{U}\mathbb{C}\mathbb{U}$, where $\mathbb{U}$ is
given in (340). Thus, ${\psi}^{(4)}$ and ${\psi}^{(7)}$ are degeneate because
they satisfy the same characteristic equation.

For $\bm{m}=\frac{3}{2}$ the two-terms series $\psi^{(1)}$, $\psi^{(4)}$,
$\psi^{(6)}$ and $\psi^{(7)}$ are all of them degenerate and each one
corresponds to two real values, ${\cal E}^{(\pm)}$, for the energy ${\cal E}$
(regardless of the Arscott condition). In fact, from the characteristic
equations
${\alpha}_{0}^{(i)}{\gamma}_{1}^{(i)}-{\beta}_{0}^{(i)}{\beta}_{1}^{(i)}=0$,
we find

$\displaystyle\begin{array}[]{l}{\cal
E}^{(\pm)}=\bm{l}^{2}+\bm{l}+\frac{5}{2}+\frac{5}{2}k^{2}\pm\sqrt{4(1-k^{2})(\bm{l}^{2}+\bm{l})+1},\qquad\bm{m}=\frac{3}{2}.\end{array}$

The even solutions, $\psi^{(1)}$ and $\psi^{(7)}$, can be written in term of
elementary functions as

$\displaystyle\begin{array}[]{l}\psi^{(1)}(u)={b}_{0}^{(1)}(\operatorname{dn}{u})^{\frac{3}{2}}\bigg{\\{}\frac{1}{\Gamma\left(1/2\right)}\left[\left(\frac{\operatorname{cn}{u}+\sqrt{k^{2}-1}\operatorname{sn}{u}}{\operatorname{dn}{u}}\right)^{\bm{l}-\frac{1}{2}}+\left(\frac{\operatorname{cn}{u}-\sqrt{k^{2}-1}\operatorname{sn}{u}}{\operatorname{dn}{u}}\right)^{\bm{l}-\frac{1}{2}}\right]+\\\
\frac{\sqrt{k^{2}-1}\quad\beta_{0}^{(1)}}{\Gamma\left(3/2\right)(1+2\bm{l})\;\alpha_{0}^{(1)}}\frac{\operatorname{sn}{u}}{\operatorname{dn}{u}}\left[\left(\frac{\operatorname{cn}{u}+\sqrt{k^{2}-1}\operatorname{sn}{u}}{\operatorname{dn}{u}}\right)^{\bm{l}+\frac{1}{2}}-\left(\frac{\operatorname{cn}{u}-\sqrt{k^{2}-1}\operatorname{sn}{u}}{\operatorname{dn}{u}}\right)^{\bm{l}+\frac{1}{2}}\right]\bigg{\\}},\quad\bm{m}=\frac{3}{2},\end{array}$
$\displaystyle\begin{array}[]{l}\psi^{(7)}(u)={b}_{0}^{(7)}(\operatorname{dn}{u})^{\frac{3}{2}}\bigg{\\{}\frac{1}{\Gamma\left(1/2\right)}\left[\left(\frac{\operatorname{cn}{u}+\sqrt{k^{2}-1}\operatorname{sn}{u}}{\operatorname{dn}{u}}\right)^{\bm{l}+\frac{3}{2}}+\left(\frac{\operatorname{cn}{u}-\sqrt{k^{2}-1}\operatorname{sn}{u}}{\operatorname{dn}{u}}\right)^{\bm{l}+\frac{3}{2}}\right]+\\\
\frac{\sqrt{k^{2}-1}\quad\beta_{0}^{(7)}}{\Gamma\left(3/2\right)(1+2\bm{l})\;\alpha_{0}^{(7)}}\frac{\operatorname{sn}{u}}{\operatorname{dn}{u}}\left[\left(\frac{\operatorname{cn}{u}+\sqrt{k^{2}-1}\operatorname{sn}{u}}{\operatorname{dn}{u}}\right)^{\bm{l}+\frac{1}{2}}-\left(\frac{\operatorname{cn}{u}-\sqrt{k^{2}-1}\operatorname{sn}{u}}{\operatorname{dn}{u}}\right)^{\bm{l}+\frac{1}{2}}\right]\bigg{\\}},\qquad\bm{m}=\frac{3}{2},\end{array}$

where there are two expressions for $\beta_{0}^{(1)}$ and $\beta_{0}^{(7)}$
corresponding to ${\cal E}={\cal E}^{(\pm)}$.

As ${{\alpha}}_{n}^{(4)}={{\alpha}}_{n}^{(7)}<0$ in (353) and (354), the
Arscott condition (82) for $\bm{m}\geq 3/2$ requires that
${{\gamma}}_{n}^{(4)}<0$ and ${{\gamma}}_{n}^{(7)}<0$ or, equivalently, that

$\displaystyle\begin{array}[]{l}F(n)=\left(n+\frac{\bm{l-m}+1}{2}\right)\left(n+\frac{\bm{l-m}}{2}\right)>0\quad\mbox{for}\quad{\psi}^{(4)},\end{array}\vspace{2mm}$
(362)
$\displaystyle\begin{array}[]{l}G(n)=\left(n-\frac{\bm{l+m}+2}{2}\right)\left(n-\frac{\bm{l+m+1}}{2}\right)>0\quad\mbox{for}\quad{\psi}^{(7)}.\end{array}$
(364)

These conditions are satisfied only for special values of $\bm{l}$ and
$\bm{m}$; for the Lamé equation ($\bm{l}=0$, $\bm{l}=-1$), we find that

$\displaystyle\begin{array}[]{l}\mbox{for}\;\bm{l}=0,\quad
F(n)>0\quad\mbox{and}\quad G(n)>0\quad\mbox{ only if
}\quad\bm{m}=\frac{3}{2},\frac{7}{2},\frac{13}{2},\cdots;\vspace{3mm}\\\
\mbox{for}\;\bm{l}=-1,\quad F(n)=G(n)>0\quad\mbox{ only if
}\quad\bm{m}=\frac{5}{2},\frac{9}{2},\frac{13}{2},\cdots.\end{array}$ (367)

By comparing (348) and (367), we see that for a given choice of
($\bm{l},\bm{m}$): (i) if the solutions ${\psi}^{(1)}$ and ${\psi}^{(6)}$
satisfy the Arscott condition, the solutions ${\psi}^{(4)}$ and ${\psi}^{(7)}$
do not satisfy; (i) if the solutions ${\psi}^{(1)}$ and ${\psi}^{(6)}$ do not
satisfy the Arscott condition, the solutions ${\psi}^{(4)}$ and ${\psi}^{(7)}$
satisfy the condition. Therefore, for the Lamé equation there are solutions
satisfying the Arscoott condition for all admissible values of
($\bm{l},\bm{m}$).

The above conclusion is not restricted to Lamé equation. Since

$\displaystyle\begin{array}[]{l}F(n)\leq 0\;\mbox{ if
}\;\frac{\bm{m-l-1}}{2}\leq n\leq\frac{\bm{m-\bm{l}}}{2},\quad\mbox{and
}\;G(n)\leq 0\;\mbox{ if }\;\frac{\bm{m+l+1}}{2}\leq
n\leq\frac{\bm{m+l+2}}{2},\end{array}$ (369)

Arscott condition for ${\psi}^{(4)}$ and ${\psi}^{(7)}$ holds if the above
intervals are empty similarly to the previous cases. As $n$ is an integer, in
the intervals (350) and (369) we can take the sign $<$ instead of $\leq$.
Thus, for ${\psi}^{(4)}$ and ${\psi}^{(1)}$ we have, respectively,

$\displaystyle\begin{array}[]{l}{\psi}^{(4)},{\psi}^{(1)}:F(n)\leq 0\;\mbox{
if }\;\frac{\bm{m-l-1}}{2}<n<\frac{\bm{m-\bm{l}}}{2},\quad\mbox{and
}\;f(n)\leq 0\;\mbox{ if
}\;\frac{\bm{m-l}}{2}<n<\frac{\bm{m-l+1}}{2}.\end{array}$

Hence, at least one of the intervals is empty for $n=2,3,4,\cdots$.
Analogously, for ${\psi}^{(6)}$ and ${\psi}^{(7)}$ we have

$\displaystyle\begin{array}[]{l}{\psi}^{(6)},{\psi}^{(7)}:g(n)\leq 0\;\mbox{
if }\;\frac{\bm{m+l}}{2}<n<\frac{\bm{m+\bm{l}}+1}{2},\quad\mbox{and
}\;G(n)\leq 0\;\mbox{ if
}\;\frac{\bm{m+l+1}}{2}<n<\frac{\bm{m+l+2}}{2}.\end{array}$

####  5.3.2 Finite series when $\bm{l}$ is an integer and $\bm{m}\neq-1/2$ is
half an odd negative integer

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
$(n-\alpha+\beta)=(n+\bm{m}+1/2)$ implies $\gamma_{n}^{(i)}=0$ for
$n=-\bm{m}-1/2$ as demanded in (75). Since $\delta=1/2$, by (178) the four
solutions converge when $z=1$ ($\mathrm{sn}^{2}u=1$).

The eigenfunctions ${\psi}^{(2)}(u)$ and ${\psi}^{(5)}(u)$ are

$\displaystyle\begin{cases}{\psi}^{(2)}(u)\stackrel{{\scriptstyle\text{(\ref{adendos-2},\ref{second-
ganguly-3})}}}{{=}}\displaystyle\frac{\mathrm{sn}\,u}{\mathrm{dn}^{\bm{m}+2}u}\sum_{n=0}^{-\bm{m}-3/2}\begin{array}[]{l}{b}_{n}^{(2)}\left[(1-k^{2})\;\mathrm{sd}^{2}u\right]^{n}\times\end{array}\\\
\tilde{F}\left[n+\frac{\bm{l}+\bm{m}+3}{2},-\frac{\bm{m}+\bm{l}+1}{2};n+\frac{3}{2};(1-k^{2})\;\mathrm{sd}^{2}u\right],\quad\bm{m}=-\frac{3}{2},-\frac{5}{2},\cdots,\;[\mbox{odd,\;
4K}],\vspace{3mm}\\\
{\alpha}_{n}^{(2)}=\left(1-\frac{1}{k^{2}}\right)(n+1),\;\;{\beta}_{n}^{(2)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\left[\bm{l}-\frac{\bm{l+m}+2}{k^{2}}\right]n+\frac{(\bm{l}+\bm{m}+2)^{2}-{\cal
E}}{4k^{2}}\vspace{3mm}\\\
-\frac{2\bm{lm}+3\bm{l}+\bm{m}^{2}+7\bm{m}+8}{4},\;\;\;{\gamma}_{n}^{(2)}=\left(n+\frac{\bm{l+m+1}}{2}\right)\left(n+\frac{\bm{l+m}+2}{2}\right)\left(n+\bm{m}+\frac{1}{2}\right);\end{cases}$
(372)
$\displaystyle\begin{cases}{\psi}^{(5)}(u)\stackrel{{\scriptstyle\text{(\ref{adendos-5},\ref{second-
ganguly-3})}}}{{=}}\displaystyle\frac{1}{\mathrm{dn}^{m+1}u}\displaystyle\sum_{n=0}^{-\bm{m}-{3}/{2}}\begin{array}[]{l}{b}_{n}^{(5)}\left[(1-k^{2})\;\mathrm{sd}^{2}u\right]^{n}\times\end{array}\vspace{2mm}\\\
\tilde{F}\left[n+\frac{\bm{m}-\bm{l}+1}{2},\frac{\bm{l}-\bm{m}-1}{2};n+\frac{1}{2};(1-k^{2})\;\mathrm{sd}^{2}u\right],\quad\bm{m}=-\frac{3}{2},-\frac{5}{2},\cdots,\;[\mbox{even,\;4K}],\vspace{3mm}\\\
{\alpha}_{n}^{(5)}=\left(1-\frac{1}{k^{2}}\right)(n+1),\quad{\beta}_{n}^{(5)}=\left(\frac{1}{k^{2}}-2\right)n^{2}-\left[\bm{l}+\frac{\bm{m}-\bm{l}+1}{k^{2}}\right]n+\frac{(\bm{l}-\bm{m}-1)^{2}-{\cal
E}}{4k^{2}}\vspace{3mm}\\\
-\frac{(\bm{l}-\bm{m}-1)^{2}+\bm{l}+\bm{m}+2-(\bm{l}+1)^{2}}{4},\qquad{\gamma}_{n}^{(5)}=\left(n+\frac{\bm{m-l}}{2}\right)\left(n+\frac{\bm{m-l}-1}{2}\right)\left(n+\bm{m}+\frac{1}{2}\right).\end{cases}$
(373)

We can verify that ${\psi}^{(2)}$ and ${\psi}^{(5)}$ result, respectively,
from ${\psi}^{(6)}$ and ${\psi}^{(1)}$ by means of the substitutions (206). By
this reason we do not discuss the degeneracy and the Arscott condition.

On the other side, the eigenfunctions ${\psi}^{(3)}(u)$ and ${\psi}^{(8)}(u)$
are

$\displaystyle\begin{cases}{\psi}^{(3)}(u)\stackrel{{\scriptstyle\text{(\ref{adendos-3},\ref{second-
ganguly-3})}}}{{=}}(\operatorname{dn}u)^{-m-1}\displaystyle\sum_{n=0}^{-\bm{m}-{3}/{2}}\begin{array}[]{l}{b}_{n}^{(3)}\left[(1-k^{2})\;\mathrm{sd}^{2}u\right]^{n}\times\end{array}\vspace{2mm}\\\
\tilde{F}\left[n+\frac{\bm{l}+\bm{m}+2}{2},-\frac{\bm{m}+\bm{l}+2}{2};n+\frac{1}{2};(1-k^{2})\;\mathrm{sd}^{2}u\right],\quad\bm{m}=\frac{1}{2},\frac{3}{2},\frac{5}{2},\quad\cdots\;[\mbox{even,\;{
4K}}],\vspace{3mm}\\\
{\alpha}_{n}^{(3)}=\left(1-\frac{1}{k^{2}}\right)(n+1),\qquad{\beta}_{n}^{(3)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\left[\left(\frac{1}{k^{2}}-1\right)(\bm{l+m}+2)-\bm{m}-1\right]n\vspace{3mm}\\\
+\frac{(\bm{l}+\bm{m}+2)^{2}-{\cal
E}}{4k^{2}}-\frac{1}{4}\left[\bm{l}+\bm{m}+2+2\bm{(l+1)(m+1)}+(\bm{m+1})^{2}\right],\vspace{3mm}\\\
{\gamma}_{n}^{(3)}=\left(n+\frac{\bm{l+m}}{2}\right)\left(n+\frac{\bm{l+m+1}}{2}\right)\left(n+\bm{m}+\frac{1}{2}\right);\end{cases}$
(374)
$\displaystyle\begin{cases}{\psi}^{(8)}(u)\stackrel{{\scriptstyle\text{(\ref{adendos-8},\ref{second-
ganguly-3})}}}{{=}}\operatorname{sn}{u}\left(\mathrm{dn}\,u\right)^{\bm{-m-2}}\displaystyle\sum_{n=0}^{-\bm{m}-3/2}\begin{array}[]{l}{b}_{n}^{(8)}\left[(1-k^{2})\;\mathrm{sd}^{2}u\right]^{n}\times\end{array}\\\
\begin{array}[]{l}\tilde{F}\left[n-\frac{\bm{l}-\bm{m}-2}{2},\frac{\bm{l}-\bm{m}}{2};n+\frac{3}{2};(1-k^{2})\;\mathrm{sd}^{2}u\right],\quad\bm{m}=-\frac{3}{2},-\frac{5}{2},-\frac{7}{2},\cdots,\quad[\mbox{odd,\;
{4K}}],\end{array}\vspace{3mm}\\\
{\alpha}_{n}^{(8)}=\left(1-\frac{1}{k^{2}}\right)(n+1),\quad{\beta}_{n}^{(8)}=\left(\frac{1}{k^{2}}-2\right)n^{2}+\left[\left(1-\frac{1}{k^{2}}\right)(\bm{l-m}-1)-\bm{m}-2\right]n+\vspace{3mm}\\\
\frac{(\bm{l}-1-\bm{m})^{2}-{\cal
E}}{4k^{2}}+\frac{2(\bm{l}+1)(\bm{m}+1)-(\bm{m}+1)^{2}-5\bm{m}-6+\bm{l}}{4},\vspace{3mm}\\\
{\gamma}_{n}^{(8)}=\left(n-\frac{\bm{l-m-1}}{2}\right)\left(n-\frac{\bm{l-m}}{2}\right)\left(n+\bm{m}+\frac{1}{2}\right).\end{cases}$
(375)

${\psi}^{(3)}$ and ${\psi}^{(8)}$ result, respectively, from ${\psi}^{(7)}$
and ${\psi}^{(4)}$ by means of the substitutions (206).

###  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

These solutions are obtained from the expansions $\bm{H}^{(i)}(x)$ given in
Section 4.3; alternatively, they can be generated from the above solutions
(Sec. 5.3) through the substitutions (207). If $\bm{l}\neq-1/2$ is half an odd
integer, then the factor $(n-\alpha-\beta+\gamma+\delta)=(n-\bm{l}-1/2)$
implies that four of the coefficients $\bm{\gamma}_{n}^{(i)}$ vanish for
$n=\bm{l}+1/2$. These give finite series with $0\leq n\leq\bm{l}-\frac{1}{2}$
if $\bm{l}$ is positive; according to (197) the expansions converge for all
values of $u$. Similarly, the factor
$(n+\alpha+\beta-\gamma-\delta)=(n+\bm{l}+1/2)$ implies that four expansions
terminate at $n=-\bm{l}-{3}/{2}$ if $\bm{l}\leq-{3}/{2}$, and by (197) the
eigenfunctions are bounded for all permissible values of $u$.

Due to the aforementioned connection with solutions of Section 5.3, the
eigenfunctions are degenerate and a given expansion satisfies the Arscott
condition (82) only for special values of $\bm{m}$ and $\bm{l}$. However, once
more, by considering all the expansions, we find at least one expansion
satisfying the condition for any allowed value of ($\bm{m}$, $\bm{l}$).

####  5.4.1 Finite series when $\bm{m}$ is an integer and $\bm{l}$ is half an
odd positive integer

Now we write the degenerate eigenfunctions $\bm{\Psi}^{(5)}(u)$ and
$\bm{\Psi}^{(7)}(u)$. By using the Eqs. (209)) we find they can be obtained
from ${\Psi}^{(1)}(u)$ and ${\Psi}^{(6)}(u)$.

$\displaystyle\begin{cases}\bm{\Psi}^{(5)}(u)\stackrel{{\scriptstyle\text{(\ref{h-bold-5},\ref{second-
ganguly-3})}}}{{=}}\displaystyle\frac{1}{\mathrm{dn}^{\bm{l}}u}\;\sum_{n=0}^{\bm{l}-1/2}\begin{array}[]{l}\bm{b}_{n}^{(5)}\;\mathrm{cn}^{2n}{u}\;\tilde{F}\left(n-\frac{\bm{l}-\bm{m}-1}{2},\frac{\bm{l}-\bm{m}-1}{2};n+\frac{1}{2};\mathrm{cn}^{2}{u}\right)\end{array}\\\
\hskip 45.52458pt\begin{array}[]{l}\stackrel{{\scriptstyle\text{(\ref{sd-
cd-5})}}}{{=}}{\Psi}^{(1)}(u+K)\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\hskip
28.45274pt\mathrm{l}=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots,\quad\left[\mbox{even,
period }4K\right]\end{array}\vspace{3mm}\\\
\begin{array}[]{l}\bm{\alpha}_{n}^{(5)}={\alpha}_{n}^{(1)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}={\alpha}_{n}^{(1)},\qquad\bm{\beta}_{n}^{(5)}={\beta}_{n}^{(1)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}\qquad\bm{\gamma}_{n}^{(5)}={\gamma}_{n}^{(1)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}};\end{array}\end{cases}$
(376)
$\displaystyle\begin{cases}\bm{\Psi}^{(7)}(u)\stackrel{{\scriptstyle\text{(\ref{h-bold-7},\ref{second-
ganguly-3})}}}{{=}}\displaystyle\frac{\mathrm{cn}\,u}{\mathrm{dn}^{\bm{l}}u}\sum_{n=0}^{\bm{l}-1/2}\begin{array}[]{l}\bm{b}_{n}^{(7)}\;\mathrm{cn}^{2n}{u}\;\tilde{F}\left(n-\frac{\bm{l}+\bm{m}-1}{2},\frac{\bm{l}+\bm{m}+1}{2};n+\frac{3}{2};\mathrm{cn}^{2}{u}\right)\end{array}\\\
\hskip 42.67912pt\begin{array}[]{l}\stackrel{{\scriptstyle\text{(\ref{sd-
cd-5})}}}{{=}}{\Psi}^{(6)}(u+K)\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\hskip
28.45274pt\mathrm{l}=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots,\quad\left[\mbox{odd
, period }4K\right]\end{array}\vspace{3mm}\\\
\begin{array}[]{l}\bm{\alpha}_{n}^{(7)}={\alpha}_{n}^{(6)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}={\alpha}_{n}^{(6)},\qquad\bm{\beta}_{n}^{(7)}={\alpha}_{n}^{(6)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\qquad\bm{\gamma}_{n}^{(7)}={\gamma}_{n}^{(6)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\end{array}.\end{cases}$
(377)

On the other side, degenerate eigenfunctions $\bm{\Psi}^{(6)}(u)$ and
$\bm{\Psi}^{(8)}(u)$ result from $\bm{H}^{(6)}(u)$ and $\bm{H}^{(8)}(u)$. By
Eqs. (209)) we see that they can also be obtained from ${\Psi}^{(7)}(u)$ and
${\Psi}^{(4)}(u)$, respectively.

$\displaystyle\begin{cases}\bm{\Psi}^{(6)}(u)\stackrel{{\scriptstyle\text{(\ref{h-bold-6},\ref{second-
ganguly-3})}}}{{=}}\displaystyle\frac{1}{\mathrm{dn}^{\bm{l}}u}\;\sum_{n=0}^{\bm{l}-1/2}\begin{array}[]{l}\bm{b}_{n}^{(6)}\;\mathrm{cn}^{2n}{u}\;\tilde{F}\left(n-\frac{\bm{l}+\bm{m}}{2},\frac{\bm{l}+\bm{m}}{2};n+\frac{1}{2};\mathrm{cn}^{2}{u}\right)\end{array}\\\
\hskip 42.67912pt\begin{array}[]{l}\stackrel{{\scriptstyle\text{(\ref{sd-
cd-5})}}}{{=}}{\Psi}^{(7)}(u+K)\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\hskip
28.45274pt\mathrm{l}=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots,\quad\left[\mbox{even,
period }4K\right]\end{array}\vspace{3mm}\\\
\begin{array}[]{l}\bm{\alpha}_{n}^{(6)}={\alpha}_{n}^{(7)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}={\alpha}_{n}^{(7)},\qquad\bm{\beta}_{n}^{(6)}={\beta}_{n}^{(7)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}\qquad\bm{\gamma}_{n}^{(6)}={\gamma}_{n}^{(7)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}};\end{array}\end{cases}$
(378)

$\displaystyle\begin{cases}\bm{\Psi}^{(8)}(u)\stackrel{{\scriptstyle\text{(\ref{h-bold-8},\ref{second-
ganguly-3})}}}{{=}}\displaystyle\frac{\operatorname{cn}{u}}{\mathrm{dn}^{\bm{l}}u}\sum_{n=0}^{\bm{l}-1/2}\begin{array}[]{l}\bm{b}_{n}^{(8)}\;\mathrm{cn}^{2n}{u}\;\tilde{F}\left(n-\frac{\bm{l}-\bm{m}-2}{2},\frac{\bm{l}-\bm{m}}{2};n+\frac{3}{2};\mathrm{cn}^{2}{u}\right)\end{array}\\\
\hskip 42.67912pt\begin{array}[]{l}\stackrel{{\scriptstyle\text{(\ref{sd-
cd-5})}}}{{=}}{\Psi}^{(4)}(u+K)\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\hskip
28.45274pt\mathrm{l}=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots,\quad\left[\mbox{odd,
period }4K\right]\end{array}\vspace{3mm}\\\
\begin{array}[]{l}\bm{\alpha}_{n}^{(8)}={\alpha}_{n}^{(4)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}={\alpha}_{n}^{(4)},\qquad\bm{\beta}_{n}^{(8)}={\alpha}_{n}^{(4)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\qquad\bm{\gamma}_{n}^{(8)}={\gamma}_{n}^{(4}\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\end{array}.\end{cases}$
(379)

For one-term series ($\bm{l}=1/2$), only two expansions are independent since

$\displaystyle\begin{array}[]{l}\bm{\psi}^{(5)}(u)=\bm{\Psi}^{(6)}(u)=\frac{\bm{b}_{0}^{(5)}}{\sqrt{\operatorname{dn}{u}}}\left[\left({\operatorname{cn}{u}-i\operatorname{sn}{u}}\right)^{\bm{m}+\frac{1}{2}}+\left({\operatorname{cn}{u}+i\operatorname{sn}{u}}\right)^{\bm{m}+\frac{1}{2}}\right],\qquad\bm{l}=\frac{1}{2},\vspace{2mm}\\\
\bm{\psi}^{(7)}(u)=\bm{\Psi}^{(8)}(u)=\frac{\bm{b}_{0}^{(7)}}{\sqrt{\operatorname{dn}{u}}}\left[\left({\operatorname{cn}{u}-i\operatorname{sn}{u}}\right)^{\bm{m}+\frac{1}{2}}-\left({\operatorname{cn}{u}+i\operatorname{sn}{u}}\right)^{\bm{m}+\frac{1}{2}}\right],\qquad\bm{l}=\frac{1}{2}.\end{array}$

These are even and odd solutions, respectively, both with period $4K$ and
energy ${\cal E}=\left(\bm{m}+\frac{1}{2}\right)^{2}+\frac{k^{2}}{4}$. We can
also verify that, for $\bm{l}=3/2$, the four two-term series are degenerate
with energies ${\cal
E}^{(\pm)}=\bm{m}^{2}+\bm{m}+\frac{5}{2}+\frac{5}{2}k^{2}\pm\sqrt{4(1-k^{2})(\bm{m}^{2}+\bm{m})+1}$.

####  5.4.2 Finite series when $\bm{m}$ is an integer and $\bm{l}\neq-1/2$ is
half an odd negative integer

For the eigenfunctions ${\bm{\Psi}}^{(1)}(u)$ and ${\bm{\Psi}}^{(3)}(u)$ we
find

$\displaystyle\begin{cases}\bm{\Psi}^{(1)}(u)\stackrel{{\scriptstyle\text{(\ref{h-bold-1},\ref{second-
ganguly-3})}}}{{=}}\displaystyle{\mathrm{dn}^{\bm{l}+1}u}\;\sum_{n=0}^{-\bm{l}-3/2}\begin{array}[]{l}\bm{b}_{n}^{(1)}\;\mathrm{cn}^{2n}{u}\;\tilde{F}\left(n+\frac{\bm{l}-\bm{m}+1}{2},\frac{\bm{m}-\bm{l}-1}{2};n+\frac{1}{2};\mathrm{cn}^{2}{u}\right)\end{array}\vspace{3mm}\\\
\begin{array}[]{l}\hskip 42.67912pt\stackrel{{\scriptstyle\text{(\ref{sd-
cd-5})}}}{{=}}{\psi}^{(5)}(u+K)\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\hskip
28.45274pt\mathrm{l}=-\frac{3}{2},-\frac{5}{2},-\frac{7}{2},\cdots,\quad\left[\mbox{even,
period }4K\right]\end{array}\vspace{3mm}\\\
\begin{array}[]{l}\bm{\alpha}_{n}^{(1)}={\alpha}_{n}^{(5)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}={\alpha}_{n}^{(5)},\qquad\bm{\beta}_{n}^{(1)}={\beta}_{n}^{(5)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}\qquad\bm{\gamma}_{n}^{(1)}={\gamma}_{n}^{(5)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}};\end{array}\end{cases}$
(381)
$\displaystyle\begin{cases}\bm{\Psi}^{(3)}(u)\stackrel{{\scriptstyle\text{(\ref{h-bold-3},\ref{second-
ganguly-3})}}}{{=}}\displaystyle{\mathrm{cn}\,u}\;{\mathrm{dn}^{\bm{l+1}}u}\sum_{n=0}^{-\bm{l}-3/2}\begin{array}[]{l}\bm{b}_{n}^{(3)}\;\mathrm{cn}^{2n}{u}\;\tilde{F}\left(n+\frac{\bm{l}+\bm{m}+3}{2},-\frac{\bm{l}+\bm{m}+1}{2};n+\frac{3}{2};\mathrm{cn}^{2}{u}\right)\end{array}\\\
\begin{array}[]{l}\hskip 42.67912pt\stackrel{{\scriptstyle\text{(\ref{sd-
cd-5})}}}{{=}}{\psi}^{(2)}(u+K)\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\hskip
28.45274pt\mathrm{l}=-\frac{3}{2},-\frac{5}{2},-\frac{7}{2},\cdots,\quad\left[\mbox{odd,
period }4K\right]\end{array}\vspace{3mm}\\\
\begin{array}[]{l}\bm{\alpha}_{n}^{(3)}={\alpha}_{n}^{(2)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}={\alpha}_{n}^{(2)},\qquad\bm{\beta}_{n}^{(3)}={\beta}_{n}^{(2)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\qquad\bm{\gamma}_{n}^{(3)}={\gamma}_{n}^{(2)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}.\end{array}\end{cases}$
(382)

${\bm{\Psi}}^{(1)}$ and ${\bm{\Psi}}^{(3)}$ follow also from
${\bm{\Psi}}^{(5)}$ and ${\bm{\Psi}}^{(7)}$ by the substitutions (206).

The degenerate eigenfunctions ($\bm{\Psi}^{(2)},\bm{\Psi}^{(4)}$), coming from
($\bm{H}^{(2)},\bm{H}^{(4)}$), are given by

$\displaystyle\begin{cases}\bm{\Psi}^{(2)}(u)\stackrel{{\scriptstyle\text{(\ref{h-bold-2},\ref{second-
ganguly-3})}}}{{=}}\displaystyle{\mathrm{dn}^{\bm{l}+1}u}\;\sum_{n=0}^{-\bm{l}-\frac{3}{2}}\begin{array}[]{l}\bm{b}_{n}^{(2)}\;\mathrm{cn}^{2n}{u}\;\tilde{F}\left(n+\frac{\bm{l}+\bm{m}+2}{2},-\frac{\bm{m}+\bm{l}+2}{2};n+\frac{1}{2};\mathrm{cn}^{2}{u}\right)\end{array}\vspace{3mm}\\\
\hskip 42.67912pt\begin{array}[]{l}\stackrel{{\scriptstyle\text{(\ref{sd-
cd-5})}}}{{=}}{\psi}^{(3)}(u+K)\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\hskip
28.45274pt\bm{l}=-\frac{3}{2},-\frac{5}{2},-\frac{7}{2},\cdots,\quad\left[\mbox{even,
period }4K\right]\end{array}\vspace{3mm}\\\
\begin{array}[]{l}\bm{\alpha}_{n}^{(2)}={\alpha}_{n}^{(3)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}={\alpha}_{n}^{(3)},\qquad\bm{\beta}_{n}^{(2)}={\beta}_{n}^{(3)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}\qquad\bm{\gamma}_{n}^{(2)}={\gamma}_{n}^{(3)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}};\end{array}\end{cases}$
(383)
$\displaystyle\begin{cases}\bm{\Psi}^{(4)}(u)\stackrel{{\scriptstyle\text{(\ref{h-bold-4},\ref{second-
ganguly-3})}}}{{=}}\displaystyle{\mathrm{cn}\,u}\;{\mathrm{dn}^{\bm{l+1}}u}\sum_{n=0}^{-\bm{l}-\frac{3}{2}}\begin{array}[]{l}\bm{b}_{n}^{(4)}\;\operatorname{cn}^{2n}{u}\;\tilde{F}\left(n+\frac{\bm{l}-\bm{m}+2}{2},\frac{\bm{m}-\bm{l}}{2};n+\frac{3}{2};\mathrm{cn}^{2}{u}\right)\end{array}\\\
\hskip 42.67912pt\begin{array}[]{l}\stackrel{{\scriptstyle\text{(\ref{sd-
cd-5})}}}{{=}}{\psi}^{(8)}(u+K)\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\hskip
22.76219pt\bm{l}=-\frac{3}{2},-\frac{5}{2},-\frac{7}{2},\cdots,\quad\left[\mbox{odd,
period }4K\right]\end{array}\vspace{3mm}\\\
\begin{array}[]{l}\bm{\alpha}_{n}^{(4)}={\alpha}_{n}^{(8)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}={\alpha}_{n}^{(8)},\qquad\bm{\beta}_{n}^{(4)}={\beta}_{n}^{(8)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}},\qquad\bm{\gamma}_{n}^{(4)}={\gamma}_{n}^{(8)}\Big{|}_{\bm{l}\leftrightarrow\bm{m}}.\end{array}\end{cases}$
(384)

Notice that ${\bm{\Psi}}^{(2)}$ and ${\bm{\Psi}}^{(4)}$ follow from
${\bm{\Psi}}^{(6)}$ and ${\bm{\Psi}}^{(8)}$ by the substitutions (206). Beside
this, for one-term series ($\bm{l}=-3/2$): $\bm{\Psi}^{(1)}=\bm{\Psi}^{(2)}$
and $\bm{\Psi}^{(3)}=\bm{\Psi}^{(4)}$.

###  5.5 Infinite-series solutions

To obtain infinite-series solutions we require that
$\mathring{\gamma}_{n}^{(i)}\neq 0$ in the power series expansions
$\mathring{H}^{(i)}(x)$ written in section 3.1. This imposes restrictions on
the parameters $\bm{l}$ and $\bm{m}$: one condition on $\bm{l-m}$ and other on
$\bm{l+m}$. Since $a=1/k^{2}>1$, the relations (108) imply that the
eigenfunctions are convergent and bounded for any value of the independent
variable, that is, for $0\leq x=\mathrm{sn}^{2}u\leq 1$. They are denoted by
$\mathring{\Phi}^{(i)}$ while the series coefficients are denoted by
$\mathring{b}_{n}^{(i)}$,

$\displaystyle\mathring{\alpha}_{n}^{(i)}\;\mathring{b}_{n+1}^{(i)}+\mathring{\beta}_{n}^{(i)}\;\mathring{b}_{n}^{(i)}+\mathring{\gamma}_{n}^{(i)}\;\mathring{b}_{n-1}^{(i)}=0,\qquad\mathring{b}_{-1}^{(i)}=0,\qquad
n\geq 0,$ (385)

$\mathring{\alpha}_{n}^{(i)}$, $\mathring{\beta}_{n}^{(i)}$ and
$\mathring{\gamma}_{n}^{(i)}$ being defined in Section 5.1. Below

  * •

we write down the conditions to have infinite series;

  * •

we show that each finite series corresponds to four infinite-series expansions
(same values for $\bm{l}$ and $\bm{m}$).

Therefore, for this family of quasi-exactly solvable potentials, for solutions
of the Schrödinger equation we have to regard both finite-series and infinite-
series wavefunctions since the infinite series are convergent and bounded for
all values of the independent variable, in opposition to the case mentioned in
[12].

By inserting the expansions $\mathring{H}^{(i)}(x)$ into (205), we get four
types of even and odd infinite-series solutions (periods $2K$ and $4K$),
namely,

$\displaystyle\begin{array}[]{r}\text{even,}\\\
2K:\end{array}\begin{cases}\begin{array}[]{l}\mathring{\Phi}^{(1)}(u)\stackrel{{\scriptstyle\text{(\ref{S1},\ref{second-
ganguly-3}})}}{{=}}\mathrm{dn}^{\bm{l}+1}{u}\displaystyle\sum_{n=0}^{\infty}\mathring{b}_{n}^{(1)}\;\mathrm{sn}^{2n}{u},\quad\mbox{if
}\begin{array}[]{l}\frac{\bm{l}-\bm{m}-1}{2}\mbox{ and
}\frac{\bm{l}+\bm{m}}{2}\end{array}\mbox{ are not}\\\ \hskip 241.84842pt\mbox{
negative integers},\end{array}\vspace{2mm}\\\
\mathring{\Phi}^{(5)}(u)\stackrel{{\scriptstyle\text{(\ref{S5},\ref{second-
ganguly-3}})}}{{=}}\displaystyle\frac{1}{\mathrm{dn}^{\bm{l}}u}\;\sum_{n=0}^{\infty}\mathring{b}_{n}^{(5)}\;\mathrm{sn}^{2n}u,\quad\mbox{if
}\begin{array}[]{l}\frac{\bm{l}-\bm{m}+1}{2}\mbox{ and
}\frac{\bm{l}+\bm{m}+2}{2}\end{array}\mbox{ are not}\\\ \hskip
241.84842pt\mbox{ positive integers};\end{cases}$ (388)
$\displaystyle\begin{array}[]{r}\text{odd,}\\\
4K:\end{array}\begin{cases}\mathring{\Phi}^{(2)}(u)\stackrel{{\scriptstyle\text{(\ref{S2},\ref{second-
ganguly-3}})}}{{=}}\mathrm{sn}{u}\;\mathrm{dn}^{\bm{l}+1}{u}\displaystyle\sum_{n=0}^{\infty}\mathring{b}_{n}^{(2)}\;\mathrm{sn}^{2n}u,\quad\mbox{if
}\begin{array}[]{l}\frac{\bm{l}-\bm{m}}{2}\mbox{ and
}\frac{\bm{l}+\bm{m}+1}{2}\mbox{ are not}\end{array}\\\ \hskip
241.84842pt\mbox{ negative integers},\vspace{2mm}\\\
\mathring{\Phi}^{(6)}(u)\stackrel{{\scriptstyle\text{(\ref{S6},\ref{second-
ganguly-3}})}}{{=}}\displaystyle\frac{\mathrm{sn}{u}}{{\rm
dn}^{\bm{l}}u}\;\sum_{n=0}^{\infty}\mathring{b}_{n}^{(6)}\;{\rm
sn}^{2n}u,\quad\mbox{if }\begin{array}[]{l}\frac{\bm{l}-\bm{m}}{2}\mbox{ and
}\frac{\bm{l}+\bm{m}+1}{2}\end{array}\mbox{ are not}\\\ \hskip
241.84842pt\mbox{ positive integers};\\\ \end{cases}$ (391)
$\displaystyle\begin{array}[]{r}\text{even, }\\\
4K:\end{array}\begin{cases}\mathring{\Phi}^{(3)}(u)\stackrel{{\scriptstyle\text{(\ref{S3},\ref{second-
ganguly-3}})}}{{=}}\mathrm{cn}{u}\;\mathrm{dn}^{\bm{l}+1}u\displaystyle\sum_{n=0}^{\infty}\mathring{b}_{n}^{(3)}\;\mathrm{sn}^{2n}u,\quad\mbox{if
}\begin{array}[]{l}\frac{\bm{l}-\bm{m}}{2}\mbox{ and
}\frac{\bm{l}+\bm{m}+1}{2}\mbox{ are not}\end{array}\\\ \hskip
241.84842pt\mbox{ negative integers},\vspace{2mm}\\\
\mathring{\Phi}^{(7)}(u)\stackrel{{\scriptstyle\text{(\ref{S7},\ref{second-
ganguly-3}})}}{{=}}\displaystyle\frac{\mathrm{cn}{u}}{{\rm
dn}^{\bm{l}}u}\;\sum_{n=0}^{\infty}\mathring{b}_{n}^{(7)}\;{\rm
sn}^{2n}u,\quad\mbox{if }\begin{array}[]{l}\frac{\bm{l}-\bm{m}}{2}\mbox{ and
}\frac{\bm{l}+\bm{m}+1}{2}\end{array}\mbox{ are not}\\\ \hskip
241.84842pt\mbox{ positive integers};\end{cases}$ (394)
$\displaystyle\begin{array}[]{r}\mbox{odd,}\\\
2K:\end{array}\begin{cases}\mathring{\Phi}^{(4)}(u)\stackrel{{\scriptstyle\text{(\ref{S4},\ref{second-
ganguly-3}})}}{{=}}\mathrm{sn}{u}\;\mathrm{cn}{u}\;\mathrm{dn}^{\bm{l}+1}u\displaystyle\sum_{n=0}^{\infty}\mathring{b}_{n}^{(4)}\;\mathrm{sn}^{2n}u,\mbox{
if }\begin{array}[]{l}\frac{\bm{l}-\bm{m}+1}{2}\mbox{ and
}\frac{\bm{l}+\bm{m}+2}{2}\mbox{ are}\end{array}\\\ \hskip
241.84842pt\mbox{not negative integers},\vspace{2mm}\\\
\mathring{\Phi}^{(8)}(u)\stackrel{{\scriptstyle\text{(\ref{S8},\ref{second-
ganguly-3}})}}{{=}}\displaystyle\frac{\mathrm{sn}{u}\;\mathrm{cn}{u}}{{\rm
dn}^{\bm{l}}u}\;\sum_{n=0}^{\infty}\mathring{b}_{n}^{(8)}\;{\rm
sn}^{2n}u,\mbox{ if }\begin{array}[]{l}\frac{\bm{l}-\bm{m}-1}{2}\mbox{ and
}\frac{\bm{l}+\bm{m}}{2}\end{array}\mbox{ are}\\\ \hskip 241.84842pt\mbox{ not
positive integers},\end{cases}$ (397)

The expansions in each pair are related to each other by
$\bm{l}\mapsto-\bm{l}-1$ and $\bm{m}\mapsto-\bm{m}-1$. From the above
conditions we will see that there are four types of infinite-series solutions
corresponding to a given finite-series solution.

First, consider the cases in which either $\bm{l}$ or $\bm{m}$ is an integer
and the other is half an odd integer, Sections 5.3 and 5.4. Then, the eight
expansions $\mathring{\Phi}^{(i)}(u)$ are given by infinite series which,
however, reduce to four expansions because Eq. (118) implies that
$\mathring{\Phi}^{(i+4)}(u)=$ $\mathring{\Phi}^{(i)}(u)$ for $i=1,2,3,4$.

As examples for finite series
$\left(\mathring{\psi}^{(i)},\tilde{\psi}^{(i)}\right)$ in terms of
$\mathrm{sn}^{2}u$, we consider the same cases given in Section 5.1. Thus, the
finite-series $\mathring{\psi}^{(1)}$, given in (246), is associated with the
four infinite-series expansions:
$\mathring{\phi}^{(2)}=\mathring{\phi}^{(6)}$,
$\mathring{\phi}^{(3)}=\mathring{\phi}^{(7)}$, $\mathring{\phi}^{(5)}$ and
$\mathring{\phi}^{(8)}$. For the second example, $\bm{m}=-1/2$. Then
$\mathring{\psi}^{(j)}=\tilde{\psi}^{(j)}$and, by using the values for
$\bm{l}$ written in (250-262), we find four infinite-series expansions
corresponding to each of the finite series according to

$\displaystyle\mathring{\psi}^{(i)}=\tilde{\psi}^{(i)}\quad\mapsto\quad\mathring{\Phi}^{(i+4)},\qquad\mathring{\psi}^{(i+4)}=\tilde{\psi}^{(i+4)}\quad\mapsto\quad\mathring{\Phi}^{(i)},\qquad$
(398)

where $i$ assumes the four values $i=1,2,3,4$ in each case. In the third
example, we take $\bm{m}=0$ for Eq. (211); thence, from the values for
$\bm{l}$ written in (266-278), we obtain the correspondence

$\displaystyle\begin{matrix}\tilde{\psi}^{(i)}\quad\mapsto\quad\mathring{\Phi}^{(i+4)},\qquad\tilde{\psi}^{(i+4)}\quad\mapsto\quad\mathring{\Phi}^{(i)},\end{matrix}$
(399)

where $i$ assumes the four values $i=1,2,3,4$ once more.

As an example of finite-series
$\left(\mathring{\bm{\Psi}}^{(i)},\tilde{\bm{\Psi}}^{(i)}\right)$ in terms of
$\mathrm{cn}^{2}u$, given in Section 5.2, we take the Lamé equation with
$\bm{l}=0$. Then, by using the parameters written in (303-321 ), we get the
following four types of infinite-series solutions corresponding each of the 16
finite series:

$\displaystyle\begin{array}[]{l}\left(\mathring{\bm{\Psi}}^{(1)},\tilde{\bm{\Psi}}^{(1)}\right),\;\left(\mathring{\bm{\Psi}}^{(4)},\tilde{\bm{\Psi}}^{(4)}\right),\;\left(\mathring{\bm{\Psi}}^{(6)},\tilde{\bm{\Psi}}^{(6)}\right),\;\left(\mathring{\bm{\Psi}}^{(7)},\tilde{\bm{\Psi}}^{(7)}\right)\;\mapsto\;\mathring{\Phi}^{(2,3,5,8)};\end{array}\vspace{3mm}$
(401)
$\displaystyle\begin{array}[]{l}\left(\mathring{\bm{\Psi}}^{(2)},\tilde{\bm{\Psi}}^{(2)}\right),\;\left(\mathring{\bm{\Psi}}^{(3)},\tilde{\bm{\Psi}}^{(3)}\right),\;\left(\mathring{\bm{\Psi}}^{(5)},\tilde{\bm{\Psi}}^{(5)}\right),\;\left(\mathring{\bm{\Psi}}^{(8)},\tilde{\bm{\Psi}}^{(8)}\right)\;\mapsto\;\mathring{\Phi}^{(1,4,6,7)}.\end{array}$
(403)

More exemples are given in Appendix C.

##  6 Conclusions

By treating the elliptic Darboux equation as an instance of Heun’s general
equation, we have applied known transformations of variables to generate new
solutions for the latter equation. In Section 3 we have established two groups
of power series expansions and, in Section 4, two groups of expansions in
series of Gauss hypergeometric functions. Each group is constituted by eight
expansions connect to each other by homotopic transformations of the dependent
variable. After this, the solutions of the Heun equation can be particularized
for Darboux equation and its special cases.

The relations among the eight expansions of each group require further study.
In Section 3 we have seen that the solutions $\mathring{U}^{(i)}$ reduce to
four expansions under certain circumstances. On the other side, Ince [33, 40]
has discussed these relations for some solutions for the particular case of
the Lamé equation.

The solutions we have considered may be used to solve the stationary
Schrödinger equation for quasi-exactly solvable (QES) potentials. For this
purpose, firstly, in section 2.1, we have seen that the Schrödinger equation
for certain elliptic potentials reduces to Darboux equations. For particular
values of the parameters, some finite-series solutions to each of such
potentials are already known in the literature [17, 20], but finite series
turn out to be systematically generated from solutions for the Heun equation
without the need of specifying the parameters. This procedure provides as well
suitable infinite-series eigenfunctions.

As an example, in Section 5 we have considered the Schrödinger equation for
the associated Lamé potential (37) which depends on the parameters $\bm{l}$
and $\bm{m}$. When either $\bm{m}-\bm{l}$ or $\bm{l}+\bm{m}$ is an integer,
the solutions $\mathring{{H}}^{(i)}(x)$ and $\mathring{\bm{H}}^{(i)}(x)$ of
the Heun equation lead to finite-series eingenfunctions given by expansions in
series of $\mathrm{sn}^{2}u$ and $\mathrm{cn}^{2}u$, respectively. To assure
real and distinct energy eingenvalues, we use the Arscott condition (82)
which, in turn, imposes further restrictions on $\bm{l}$ and $\bm{m}$ (such
restrictions are written on the right-hand side of the expansions given in
Sections 5.1 and 5.2). 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. This fact gives a criterion
to select the appropriate solutions for the associated Lamé equation, as in
the examples of of Sections 5.1, 5.2 and Appendix C, where the use of each of
the preceding finite-series expansions has been illustrated. From the
solutions of Sections 5.1 and 5.2 we can generate new solutions by means of
the substitutions (207).

On the other hand, when either $\bm{l}$ or $\bm{m}$ is an integer and the
other is half an odd integer, we have found eigenfunctions by using the
hypergeometric-function expansions ${H}^{(i)}(x)$ and ${\bm{H}}^{(i)}(x)$
given in Section 4. If $\bm{l}$ or $\bm{m}$ is a positive or a negative
integer all the expansions afford finite-series solutions bounded for all
values of the independent variable $u$. Thus, if $\bm{l}$ is an integer and
$\bm{m}\neq-1/2$ is half an odd integer we have the degenerate finite-series
solutions given in Section 5.3; for $\bm{l}=-1$ or $\bm{l}=0$, we get new
solutions for the Lamé equation. If $\bm{m}$ is an integer and
$\bm{l}\neq-1/2$ is half an odd integer, we have the degenerate finite-series
solutions given in Section 5.4. In fact, the solutions of Sections 5.3 and 5.4
are connected by means of the substitutions (207). The eigenfunctions are
degenerate in the sense that there are two independent eigenfunctions
corresponding to the same energy. The degeneracy has been shown by writing the
recurrence relations in the matrix form.

However, Khare and Sukhatme [19, 20] have pointed out another type of
degeneracy which is not obtained by using expansions in series of
hypergeometric functions. In fact, this type of degenerate solutions comes
from expansions in series of $\mathrm{cn}^{2}u$ or $\mathrm{sn}^{2}u$ for
particular values for $\bm{l}$ and $\bm{m}$, as explained in Appendix C.

The previous examples show how to use the solutions of the Heun equation to
find finite-series solutions for the elliptic Darboux equation or, more
precisely, for the associated Lamé equation. Furthermore, to each finite-
series solution, in Section 5.5 we have seen that this procedure also gives
four periodic infinite-series solutions which are bounded and convergent for
any value of the independent variable. Hitherto, relations among these have
not been determined even for the known case of the Lamé equation [40]. We note
that the existence of such periodic solutions for the associated Lamé equation
can also be studied by using expansions in series of trigonometric functions
like the ones considered in References [40, 41, 42]. In Appendix D we explain
how these solutions could be obtained as particular instances of solutions of
the Heun equation (2).

Anyway, in the present case, infinite series as solutions for quasi-exactly
solvable problems cannot be discarded by asserting that the wavefunctions are
unbounded, as sometimes occurs [12]. Besides this, it is known that the
infinite series are important in band theory of solids. Indeed, these are the
only type of eigenstates admitted by a model proposed by Slater in which the
Schrödinger equation reduces to a Mathieu equation [37] (this does not admit
finite-series solutions).

Although we have considered only the associated Lamé potential, the solutions
of the Heun equation can also be applied for the other potentials presented in
Section 2.1. To finalize, we mention some supplementary issues. In the first
place, for the Heun equation, there are other expansions in series of
hypergeometric functions [13, 14, 15, 43] which, however, are not appropriate
for the problems of Section 2.1. The same is true respecting a recent study on
Darboux equation [44].

In the second place, one of the Erdélyi expansions in series of hypergeometric
functions seems to constitute a suitable start-point for generating new
solutions for the associated Lamé equation. To this end, from the inception we
have to modify the recurrence relations for series coefficients as in Appendix
D and, after that, use the homotopic transformations as well as the fractional
substitutions (62-66). Similarly, by using modified recurrence relations we
see that the Svartholm solution [13] leads to expansions in series of
trigonometric functions for the associated Lamé equation. However, in order to
compare solutions obtained from Svartholm’s and Erdelyi’s expansions with the
ones presented in this article, it would be necessary to find out if the
finite-series expansions fulfill the Arscott condition (82).

## Acknowledgement

This work was funded by Ministério da Ciência, Tecnologia e Inovações of
Brazil. The author thanks Léa Jaccoud El-Jaick for computations of section 4.1
and for reading the manuscript. This work does not have any conflicts of
interest.

##  Appendix A Homotopic transformations

The 8 homotopic transformations (2.3) are:

$\displaystyle\begin{array}[]{l}T_{1}H(x)=H(x)=H(a,q;\alpha,\beta,\gamma,\delta;x),\quad(\text{identity})\end{array}\vspace{2mm}$
(A.2)
$\displaystyle\begin{array}[]{l}T_{2}H(x)=x^{1-\gamma}H\big{[}a,q-(\gamma-1)(\delta
a+\epsilon);{\beta-\gamma+1},\alpha-\gamma+1,2-\gamma,\delta;x\big{]},\end{array}\vspace{2mm}$
(A.4)
$\displaystyle\begin{array}[]{l}T_{3}H(x)=(1-x)^{1-\delta}H\big{[}a,q-(\delta-1)\gamma
a;{\beta-\delta+1},\alpha-\delta+1,\gamma,2-\delta;x\big{]},\end{array}\vspace{2mm}$
(A.6)
$\displaystyle\begin{array}[]{l}T_{4}H(x)=x^{1-\gamma}(1-x)^{1-\delta}\times\vspace{2mm}\\\
H\left[a,q-(\gamma+\delta-2)a-(\gamma-1)\epsilon;{\alpha-\gamma-\delta+2},\beta-\gamma-\delta+2,2-\gamma,2-\delta;x\right],\end{array}$
(A.9)
$\displaystyle\begin{array}[]{l}T_{5}H(x)=\left[1-\frac{x}{a}\right]^{1-\epsilon}H\big{[}a,q-\gamma(\alpha+\beta-\gamma-\delta);{-\alpha+\gamma+\delta},-\beta+\gamma+\delta,\gamma,\delta;x\big{]},\end{array}\vspace{2mm}$
(A.11)
$\displaystyle\begin{array}[]{l}T_{6}H(x)=\left[1-\frac{x}{a}\right]^{1-\epsilon}x^{1-\gamma}\times\vspace{2mm}\\\
\hskip
22.76228ptH\big{[}a,q-\delta(\gamma-1)a-\alpha-\beta+\delta+1;{-\beta+\delta+1},-\alpha+\delta+1,2-\gamma,\delta;x\big{]},\end{array}\vspace{2mm}$
(A.14)
$\displaystyle\begin{array}[]{l}T_{7}H(x)=\left[1-\frac{x}{a}\right]^{1-\epsilon}(1-x)^{1-\delta}\times\vspace{2mm}\\\
\hskip
11.38092ptH\Big{[}a,q-\gamma[(\delta-1)a+\alpha+\beta-\gamma-\delta];{-\beta+\gamma+1},-\alpha+\gamma+1,\gamma,2-\delta;x\Big{]},\end{array}\vspace{2mm}$
(A.17)
$\displaystyle\begin{array}[]{l}T_{8}H(x)=\left[1-\frac{x}{a}\right]^{1-\epsilon}\displaystyle
x^{1-\gamma}(1-x)^{1-\delta}\times\vspace{2mm}\\\ \hskip
28.45274ptH\big{[}a,q-(\gamma+\delta-2)a-\alpha-\beta+\delta+1;{2-\alpha},2-\beta,2-\gamma,2-\delta;x\big{]}.\end{array}$
(A.20)

##  Appendix B Cases solved by hypergeometric functions

In the four groups of solutions there are cases where the recurrence relations
for the series coefficients present only two terms. For such cases, by
combining results given in Ref. [8] with transformations of Maier’s table [5],
we find that the Heun equation (2) is reduced to the hypergeometric equation
(4).

In (117) we have seen that the powers expansions (98-105) present two-term
recurrence relations in the following three cases:

$\displaystyle\begin{array}[]{l}a=0;\quad a=-1,\quad\epsilon=\delta,\quad
q=0\quad\mbox{and}\quad a=-1,\quad\alpha+\beta=1+\gamma,\quad
q=\gamma(1-\delta).\end{array}$

For $a=0$ the Heun equation equation reads

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

and the substitution

$\displaystyle H(x)=x^{k_{1}}F(x),\qquad
k_{1}=\frac{1}{2}\left[1-\gamma-\epsilon-\sqrt{(1-\gamma-\epsilon)^{2}-4q}\right]$

gives the hypergeometric equation

$\displaystyle
x(1-x)\frac{d^{2}F}{dx^{2}}+\big{[}(2k_{1}+\gamma+\epsilon)-(2k_{1}+\alpha+\beta+1)x\big{]}\frac{dF}{dx}-(k_{1}+\alpha)(k_{1}+\beta)F=0.$

For the second case ($a=-1,\;\epsilon=\delta,\;q=0$), the Heun equation (2)
reads

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

since $\epsilon=\delta=(\alpha+\beta+1-\delta)/2$. It is already known that
the substitutions [8]

$\displaystyle z=x^{2},\qquad u(z)=H(x),\quad[a=-1,\;\epsilon=\delta,\;q=0]$
(B.3)

reduce the above equation to

$\displaystyle
z(1-z)\frac{d^{2}u(z)}{dz^{2}}+\left[\frac{1}{2}(\gamma+1)-\frac{1}{2}(\alpha+\beta+2)z\right]\frac{du(z)}{dz}-\frac{\alpha\beta}{4}\;u(z)=0,$
(B.4)

which is the hypergeometric equation (4) with $\mathrm{a}=\alpha/2$,
$\mathrm{b}=\beta/2$ and $\mathrm{c}=(1+\gamma)/2$. For the third case, first
we transform the Heun equation (2) by using the homotopic substitution
$T_{3}$, that is, we take

$\displaystyle
H(x)=(1-x)^{1-\delta}h(x),\qquad[a=-1,\;\gamma=\alpha+\beta-1,\;q=\gamma(1-\delta)].$

Then, the Heun equation leads to

$\displaystyle\frac{d^{2}h(x)}{dx^{2}}+\left[\frac{\alpha+\beta-1}{x}+\frac{2-\delta}{x-1}+\frac{2-\delta}{x+1}\right]\frac{dh(x)}{dx}+\frac{(\alpha-\delta+1)(\beta-\delta+1)}{(x-1)(x+1)}h(x)=0.$

Now, by the substitutions [8]

$\displaystyle\begin{array}[]{r}z=x^{2},\qquad u(z)=h(x)\quad\Rightarrow\quad
H(x)=(1-x)^{1-\delta}u(z),\end{array}\vspace{2mm}$ (B.6)
$\displaystyle\left[a=-1,\;\gamma=\alpha+\beta-1,\;q=\gamma-\gamma\delta\right]]$

the above equation gives the hypergeometric equation

$\displaystyle\begin{array}[]{l}z(1-z)\frac{d^{2}u(z)}{dz^{2}}+\left[\frac{1}{2}(\alpha+\beta)-\frac{1}{2}(\alpha+\beta+4-2\delta)z\right]\frac{du(z)}{dz}-\frac{(\alpha-\delta+1)(\beta-\delta+1)}{4}\;u(z)=0,\end{array}$
(B.8)

with parameters $\mathrm{a}=(\alpha-\delta+1)/2$,
$\mathrm{b}=(\beta-\delta+1)/2$ and $\mathrm{c}=(\alpha+\beta)/2$.

According to (137) and (184) the expansions $\mathring{\bm{H}}^{(i)}(x)$ and
$H^{(i)}(x)$ have two-term recurrence relations if

$\displaystyle\begin{array}[]{l}a=1;\quad
a=2,\quad\alpha+\beta+1=2\gamma+\delta,\quad
q=\alpha\beta\quad\mbox{and}\vspace{2mm}\\\
a=2,\quad\alpha+\beta=1+\delta,\quad
q=\alpha\beta+(\gamma-1)\delta.\end{array}$

For $a=1$ we have

$\displaystyle
x(x-1)\frac{d^{2}H}{dx^{2}}+\big{[}(\alpha+\beta+1)x-\gamma\big{]}\frac{dH}{dx}+\frac{\alpha\beta
x-q}{x-1}H=0,\qquad(a=1)$ (B.10)

and the change of variable

$\displaystyle H(x)=(x-1)^{k_{2}}G(x),\qquad
k_{2}=\frac{1}{2}\left[\gamma-\alpha-\beta-\sqrt{(\gamma-\alpha-\beta)^{2}-4\alpha\beta+4q}\right]$
(B.11)

yields an hypergeometric equation, namely,

$\displaystyle
x(1-x)\frac{d^{2}G}{dx^{2}}+\big{[}\gamma-(2k_{2}+\alpha+\beta+1)x\big{]}\frac{dG}{dx}-(k_{2}+\alpha)(k_{2}+\beta)G=0.$
(B.12)

For the second case the Heun equation (2) becomes

$\displaystyle\frac{d^{2}H(x)}{dx^{2}}+\left[\frac{\gamma}{x}+\frac{\alpha+\beta+1-2\gamma}{x-1}+\frac{\gamma}{x-2}\right]\frac{dH(x)}{dx}+\frac{\alpha\beta}{x(x-2)}H(x)=0,\qquad\vspace{3mm}$
$\displaystyle\left[a=2,\;\alpha+\beta=2\gamma+\delta,\;q=\alpha\beta\right].$

It is known that the substitutions [8]

$\displaystyle\begin{array}[]{r}z=x(2-x),\qquad
u(z)=H(x),\quad\left[a=2,\;\alpha+\beta=2\gamma+\delta,\;q=\alpha\beta\right]\end{array}$
(B.14)

transform the above equation into the hypergeometric equation (4)

$\displaystyle\begin{array}[]{l}z(1-z)\frac{d^{2}u(z)}{dz^{2}}+\left[\gamma-\frac{1}{2}(\alpha+\beta+2)z\right]\frac{du(z)}{dz}-\frac{\alpha\beta}{4}\;u(z)=0,\end{array}$
(B.16)

with $\mathrm{a}=\alpha/2$, $\mathrm{b}=\beta/2$ and $\mathrm{c}=\gamma$. For
the third case, first we transform the Heun equation (2) by using the
homotopic substitution $T_{2}$, that is, we take

$\displaystyle\begin{array}[]{l}H(x)=x^{1-\gamma}h(x),\qquad[a=2,\;\alpha+\beta=1+\delta,\;q=\alpha\beta+(\gamma-1)\delta].\end{array}$

Then, the Heun equation leads to

$\displaystyle\begin{array}[]{l}\frac{d^{2}h(x)}{dx^{2}}+\left[\frac{2-\gamma}{x}+\frac{\delta}{x-1}+\frac{2-\gamma}{x-2}\right]\frac{dh(x)}{dx}+\frac{(\alpha-\gamma+1)(\beta-\gamma+1)}{x(x-2)}\;h(x)=0.\end{array}$

By the substitutions [8]

$\displaystyle\begin{array}[]{r}z=x(2-x),\qquad u(z)=h(x)\quad\Rightarrow\quad
H(x)=x^{1-\gamma}u(z),\end{array}\vspace{2mm}$ (B.20)
$\displaystyle\left[a=2,\;\alpha+\beta=1+\delta,\;q=\alpha\beta+(\gamma-1)\delta\right]$

we get the hypergeometric equation

$\displaystyle\begin{array}[]{l}z(1-z)\frac{d^{2}u(z)}{dz^{2}}+\left[2-\gamma-\frac{1}{2}(\alpha+\beta+2-2\gamma)z\right]\frac{du(z)}{dz}-\frac{(\alpha-\gamma+1)(\beta-\gamma+1)}{4}\;u(z)=0,\textasciidieresis\end{array}$
(B.22)

with $\mathrm{a}=(\alpha-\gamma+1)/2$, $\mathrm{b}=(\beta-\gamma+1)/2$ and
$\mathrm{c}=2-\gamma$.

Finally, by (201) the expansions $\bm{H}^{(i)}(x)$ present two-term recurrence
relations if

$\displaystyle\begin{array}[]{l}a=2,\quad\beta=\alpha+1-\delta,\quad
q=\alpha\gamma\quad\mbox{and}\quad a=2,\quad\alpha=\beta+1-\delta,\quad
q=\beta\gamma.\end{array}$

For the first case, in Eq. (2) we use the fractional substitution

$\displaystyle\begin{array}[]{l}H(x)=\left(1-\frac{x}{a}\right)^{-\alpha}f(y),\qquad
y=\frac{(1-a)x}{x-a},\qquad[a=2,\;\beta=\alpha+1-\delta,\;q=\alpha\gamma]\end{array}$

which is Maier the transformation $M_{17}$. Thence

$\displaystyle\frac{d^{2}f(y)}{dy^{2}}+\left[\frac{\gamma}{y}+\frac{\delta}{y-1}+\frac{\delta}{y+1}\right]\frac{df(y)}{dy}-\frac{\alpha(1+\alpha-\gamma-2\delta)}{(y-1)(y+1)}f(y)=0.$

The changes

$\displaystyle\begin{array}[]{r}z=y^{2}=\left(\frac{x}{2-x}\right)^{2},\quad
u(z)=f(y)\quad\Rightarrow\quad
H(x)=\left(1-\frac{x}{2}\right)^{-\alpha}u(z),\end{array}\vspace{2mm}$ (B.26)
$\displaystyle\left[a=2,\;\beta=\alpha+1-\delta,\;q=\alpha\gamma\right]$

give the hypergeometric equation

$\displaystyle\begin{array}[]{l}z(1-z)\frac{d^{2}u(z)}{dz^{2}}+\left[\frac{1+\gamma}{2}-\frac{1+\gamma+2\delta}{2}z\right]\frac{du(z)}{dz}+\frac{\alpha(1+\alpha-\gamma-2\delta)}{4}\;u(z)=0,\textasciidieresis\end{array}$
(B.28)

whose parameters are $\mathrm{a}=\alpha/2$,
$\mathrm{b}=(\gamma+2\delta-\alpha-1)/2$ and $\mathrm{c}=(1+\gamma)/2$. The
second case differs from the first one by the interchange
$\alpha\longleftrightarrow\beta$. Thus, the changes

$\displaystyle\begin{array}[]{r}z=y^{2}=\left(\frac{x}{2-x}\right)^{2},\;u(z)=f(y)\;\Rightarrow\;H(x)=\left(1-\frac{x}{2}\right)^{-\beta}u(z),\end{array}\vspace{2mm}$
(B.30) $\displaystyle\left[a=2,\;\alpha=\beta+1-\delta,\;q=\beta\gamma\right]$

give the hypergeometric equation

$\displaystyle\begin{array}[]{l}z(1-z)\frac{d^{2}u(z)}{dz^{2}}+\left[\frac{1+\gamma}{2}-\frac{1+\gamma+2\delta}{2}z\right]\frac{du(z)}{dz}+\frac{\beta(1+\beta-\gamma-2\delta)}{4}\;u(z)=0,\textasciidieresis\end{array}$
(B.32)

whose parameters are $\mathrm{a}=\beta/2$,
$\mathrm{b}=(\gamma+2\delta-\beta-1)/2$ and $\mathrm{c}=(1+\gamma)/2$.

##  Appendix C Some degenerate power-series solutions

We consider three cases of the Schrödinger equation (202) whose solutions are
obtained as particular cases of the power-series solutions given in Section 5.
For each case, we will find

  * •

degenerate finite-series eigenfunctions which satisfy the Arscott criterion;

  * •

four infinite-series eigenfunctions.

The cases are: (1) $\bm{l}=1/2$ and $\bm{m}=3/2$, (2) $\bm{l}=1/2$ and
$\bm{m}=5/2$, (3) $\bm{l}=1/2$ and $\bm{m}=7/2$. For these cases the finite
series have at most three terms whose coefficients are computed from the
recurrence relations (72) by means of

$\displaystyle b_{1}=-b_{0}\;\frac{\beta_{0}}{\alpha_{0}};\qquad
b_{2}=b_{0}\;\frac{1}{\alpha_{1}}\left(\frac{\beta_{0}\beta_{1}}{\alpha_{0}}-\gamma_{1}\right),$
(C.1)

where $b_{0}$ is the coefficient of the first term. The characteristic
equations (81) reduces to

$\displaystyle\beta_{0}=0,\qquad\beta_{0}\;\beta_{1}-\alpha_{0}\;\gamma_{1}=0,\qquad\beta_{0}\;\beta_{1}\;\beta_{2}-\alpha_{1}\;\beta_{0}\;\gamma_{2}-\alpha_{0}\;\beta_{2}\;\gamma_{1}=0,$
(C.2)

for series with one, two and three terms respectively.

Khare and Sukhatme [19, 20] have found solutions for these cases by
integrating the Schrödinger equation directly. Here, the solutions are
obtained from the expansions in series of $\mathrm{cn}^{2}u$
($\mathring{\bm{\Psi}}^{(i)}$ and $\tilde{\bm{\Psi}}^{(i)}$ written in Section
5.2); these satisfy the Arscott condition for series with two or more terms,
in opposition to $\mathring{\psi}^{(i)}$ and $\tilde{\psi}^{(i)}$.

### First case: $\bm{l}=1/2$ and $\bm{m}=3/2$

For this case, $\bm{m}-\bm{l}=1$ and $\bm{m}+\bm{l}=2$. Thence, (281a-288a)
imply the

$\displaystyle\mbox{finite-series
expansions}:\;\mathring{\bm{\Psi}}^{(1)}\;(1\mbox{
term}),\quad\tilde{\bm{\Psi}}^{(5)}\;(2\mbox{
terms}),\quad\tilde{\bm{\Psi}}^{(8)}\;(1\mbox{ term}),$ (C.3)

while (388-397) give four

$\displaystyle\mbox{infinite-series
expansions}:\;\mathring{\Phi}^{(2)}\stackrel{{\scriptstyle\text{(\ref{ince})}}}{{=}}\mathring{\Phi}^{(6)},\quad\mathring{\Phi}^{(3)}\stackrel{{\scriptstyle\text{(\ref{ince})}}}{{=}}\mathring{\Phi}^{(7)},\quad\mathring{\Phi}^{(4)},\quad\mathring{\Phi}^{(5)}.$
(C.4)

The one-term expansions and their respective energies read

$\displaystyle\begin{array}[]{l}\mathring{\bm{\Psi}}^{(1)}(u)\stackrel{{\scriptstyle\text{(\ref{ps1-C}})}}{{=}}\bm{b}_{0}^{(1)}\;\mathrm{dn}^{3/2}u,\hskip
28.45274pt{\cal E}=\frac{9}{4}k^{2};\end{array}\vspace{2mm}$ (C.6)
$\displaystyle\begin{array}[]{l}\tilde{\bm{\Psi}}^{(8)}(u)\stackrel{{\scriptstyle\text{(\ref{ps2-D}})}}{{=}}\bm{b}_{0}^{(8)}\;\frac{\mathrm{sn}u\;\mathrm{cn}u}{\mathrm{dn}^{1/2}u},\;\;\qquad{\cal
E}=4+\frac{1}{4}k^{2}.\end{array}$ (C.8)

The two-term expansion $\mathring{\bm{\Psi}}^{(5)}(u)$ represents two
solutions, but one is equivalent to the previous
$\mathring{\bm{\Psi}}^{(1)}(u)$, that is,

$\displaystyle\begin{array}[]{l}\tilde{\bm{\Psi}}^{(5)}_{\;1}(u)\stackrel{{\scriptstyle\text{(\ref{ps5-D}})}}{{=}}\frac{C_{1}}{\mathrm{dn}^{1/2}u}\left(1+\frac{k^{2}}{1-k^{2}}\mathrm{cn}^{2}u\right)=\frac{C_{1}}{(1-k^{2})\mathrm{dn}^{1/2}u}\left(1-k^{2}\mathrm{sn}^{2}u\right)\stackrel{{\scriptstyle\text{(\ref{imply}})}}{{=}}\mathring{\bm{\Psi}}^{(1)}(u),\end{array}\vspace{2mm}$
(C.10)
$\displaystyle\begin{array}[]{l}\tilde{\bm{\Psi}}^{(5)}_{\;2}(u)\stackrel{{\scriptstyle\text{(\ref{ps5-D}})}}{{=}}\frac{C_{2}}{\mathrm{dn}^{1/2}u}\left(1-2\mathrm{cn}^{2}u\right),\hskip
42.67912pt{\cal E}=4+\frac{1}{4}k^{2}.\end{array}$ (C.12)

Therefore, there are three independent eigensolutions.
$\tilde{\bm{\Psi}}^{(8)}(u)$ and $\tilde{\bm{\Psi}}^{(5)}_{\;2}(u)$ are
degenerate because have the same energy.

### Second case: $\bm{l}=1/2$ and $\bm{m}=5/2$

Now $\bm{m}-\bm{l}=2$ and $\bm{l}+\bm{l}=3$. Then, proceeding as in the above
case, we obtain the

$\displaystyle\begin{array}[]{l}\mbox{finite
series}:\;\mathring{\bm{\Psi}}^{(2)}\mbox{ and
}\mathring{\bm{\Psi}}^{(3)}\;(1\mbox{ term}),\;\tilde{\bm{\Psi}}^{(6)}\mbox{
and }\tilde{\bm{\Psi}}^{(7)}\;(2\mbox{ terms});\end{array}\vspace{3mm}$ (C.14)
$\displaystyle\begin{array}[]{l}\mbox{infinite series
}:\;\mathring{\Phi}^{(1)},\quad\mathring{\Phi}^{(4)},\quad\mathring{\Phi}^{(5)},\quad\mathring{\Phi}^{(8)}.\end{array}$
(C.16)

The one-term finite series give two eigenfunctions,

$\displaystyle\displaystyle\mathring{\bm{\Psi}}^{(2)}(u)\stackrel{{\scriptstyle\text{(\ref{ps2-C}})}}{{=}}\bm{b}_{0}^{(2)}\;\mathrm{sn}u\;\mathrm{dn}^{3/2}u,\hskip
22.76219pt{\cal E}=1+\frac{25}{4}k^{2};\qquad\vspace{2mm}$ (C.17)
$\displaystyle\displaystyle\mathring{\bm{\Psi}}^{(3)}(u)\stackrel{{\scriptstyle\text{(\ref{ps3-C}})}}{{=}}\bm{b}_{0}^{(3)}\;\mathrm{cn}u\;{\mathrm{dn}^{3/2}u},\qquad{\cal
E}=1+\frac{9}{4}k^{2}.$ (C.18)

The two-term series $\tilde{\bm{\Psi}}^{(6)}(u)$ represents two eigenfunctions
but one of them repeats the above solution $\mathring{\bm{\Psi}}^{(2)}(u)$,

$\displaystyle\tilde{\bm{\Psi}}^{(6)}_{\;1}(u)\stackrel{{\scriptstyle\text{(\ref{ps6-D}})}}{{=}}C_{1}\;\frac{\mathrm{sn}u}{\mathrm{dn}^{1/2}u}\left(1+\frac{k^{2}}{1-k^{2}}\mathrm{cn}^{2}u\right)=\mathring{\bm{\Psi}}^{(2)}(u),$
$\displaystyle\tilde{\bm{\Psi}}^{(6)}_{\;2}(u)\stackrel{{\scriptstyle\text{(\ref{ps6-D}})}}{{=}}C_{2}\;\frac{\mathrm{sn}u}{\mathrm{dn}^{1/2}u}\left(1-4\mathrm{cn}^{2}u\right),\qquad{\cal
E}=9+\frac{1}{4}k^{2}.$ (C.19)

Similarly, the two-term series $\tilde{\bm{\Psi}}^{(7)}(u)$ represent two
eigenfunctions but one of them repeats the above solution
$\mathring{\bm{\Psi}}^{(3)}(u)$,

$\displaystyle\begin{array}[]{l}\displaystyle\tilde{\bm{\Psi}}^{(7)}_{\;1}(u)\stackrel{{\scriptstyle\text{(\ref{ps7-D}})}}{{=}}D_{1}\;\frac{\mathrm{cn}u}{\mathrm{dn}^{1/2}u}\left(1+\frac{k^{2}}{1-k^{2}}\mathrm{cn}^{2}u\right)=\displaystyle\mathring{\bm{\Psi}}^{(3)}(u),\end{array}\vspace{
3mm}$ (C.21)
$\displaystyle\displaystyle\begin{array}[]{l}\tilde{\bm{\Psi}}^{(7)}_{\;2}(u)\stackrel{{\scriptstyle\text{(\ref{ps7-D}})}}{{=}}D_{2}\;\frac{\mathrm{cn}u}{\mathrm{dn}^{1/2}u}\left(1-\frac{4}{3}\mathrm{cn}^{2}u\right),\qquad{\cal
E}=9+\frac{1}{4}k^{2}\mbox{ (degenerate ver acima)};.\end{array}$ (C.23)

Therefore, in this case there are only four independent solutions, being
$\tilde{\bm{\Psi}}^{(6)}_{\;2}(u)$ and $\tilde{\bm{\Psi}}^{(7)}_{\;2}(u)$
degenerate solutions.

### Third case: $\bm{l}=1/2$ and $\bm{m}=7/2$

In this case $\bm{m}-\bm{l}=3$ and $\bm{m}+\bm{l}=4$ . Thence we get the the
following finite and infinite series:

$\displaystyle\mbox{finite}:\;\mathring{\bm{\Psi}}^{(1)}\;(2\mbox{
terms}),\;\mathring{\bm{\Psi}}^{(4)}\;(1\mbox{
term}),\;\tilde{\bm{\Psi}}^{(5)}\;(3\mbox{
terms}),\;\tilde{\bm{\Psi}}^{(8)}\;(2\mbox{ terms});\qquad\vspace{3mm}$ (C.24)
$\displaystyle\mbox{infinite}:\;\mathring{\Phi}^{(2)},\quad\mathring{\Phi}^{(3)},\quad\mathring{\Phi}^{(6)},\quad\mathring{\Phi}^{(7)}.$
(C.25)

The one-term series reads

$\displaystyle\begin{array}[]{l}\displaystyle\mathring{\bm{\Psi}}^{(4)}(u)\stackrel{{\scriptstyle\text{(\ref{ps4-C}})}}{{=}}\bm{b}_{0}^{(4)}\;\mathrm{sn}u\;\mathrm{cn}u\;\mathrm{dn}^{3/2}u,\hskip
28.45274pt{\cal E}=4+\frac{25}{4}k^{2}.\end{array}$ (C.27)

The two-term expansions $\mathring{\bm{\Psi}}^{(1)}(u)$ represent two
eigenfunctions $\mathring{\bm{\Psi}}^{(1)}_{\;\pm}(u)$ corresponding to the
energies ${\cal E}_{\pm}$,

$\displaystyle\begin{array}[]{l}\mathring{\bm{\Psi}}^{(1)}_{\;\pm}(u)\stackrel{{\scriptstyle\text{(\ref{ps1-C}})}}{{=}}C_{\pm}\;{\mathrm{dn}^{3/2}u}\left[1+\frac{7k^{2}-2\pm\sqrt{4+25k^{4}-4k^{2}}}{2(1-k^{2})}\;\mathrm{cn}^{2}u\right],\end{array}\quad\vspace{3mm}$
(C.29) $\displaystyle\begin{array}[]{l}\hskip 85.35826pt{\cal
E}_{\pm}=2+\frac{29}{4}k^{2}\mp\sqrt{4+25k^{4}-4k^{2}}.\end{array}$

The expansion $\mathring{\bm{\Psi}}^{(8)}(u)$ also represents two
eigenfunctions but one of these is identical to
$\mathring{\bm{\Psi}}^{(4)}(u)$. We find

$\displaystyle\begin{array}[]{l}\displaystyle\tilde{\bm{\Psi}}^{(8)}_{\;1}(u)\stackrel{{\scriptstyle\text{(\ref{ps8-D}})}}{{=}}D_{1}\;\frac{\mathrm{sn}u\;\mathrm{cn}u}{\mathrm{dn}^{1/2}u}\left[1+\frac{k^{2}}{1-k^{2}}\mathrm{cn}^{2}u\right]=\displaystyle\mathring{\bm{\Psi}}^{(4)}(u),\end{array}\vspace{
3mm}$ (C.32)
$\displaystyle\begin{array}[]{l}\displaystyle\tilde{\bm{\Psi}}^{(8)}_{\;2}(u)\stackrel{{\scriptstyle\text{(\ref{ps8-D}})}}{{=}}D_{2}\;\frac{\mathrm{sn}u\;\mathrm{cn}u}{\mathrm{dn}^{1/2}u}\left[1-2\mathrm{cn}^{2}u\right],\qquad{\cal
E}=16+\frac{1}{4}k^{2}.\end{array}$ (C.34)

On the other hand, as $\mathring{\bm{\Psi}}^{(4)}(u)$ is a three-term
expansion, it represents three eigensolutions having the form

$\displaystyle\begin{array}[]{l}\displaystyle\tilde{\bm{\Psi}}^{(5)}(u)\stackrel{{\scriptstyle\text{(\ref{ps5-D}})}}{{=}}\frac{\tilde{\bm{b}}_{\;0}^{(5)}}{\mathrm{dn}^{1/2}u}\left\\{1-\frac{2k^{2}}{k^{2}-1}\;\tilde{\bm{\beta}}_{0}^{(5)}\;\mathrm{cn}^{2}u+\right.\end{array}$
(C.36) $\displaystyle\hskip
62.59596pt\begin{array}[]{l}\left.\frac{k^{2}}{3(k^{2}-1)}\left[4+\frac{2k^{2}}{(k^{2}-1)}\;\tilde{\bm{\beta}}_{0}^{(5)}\left(\tilde{\bm{\beta}}_{0}^{(5)}+\frac{1}{k^{2}}-\frac{3}{2}\right)\right]\mathrm{cn}^{4}u\right\\},\end{array}$
(C.38)

where $\tilde{\bm{\beta}}_{0}^{(5)}$ is determined from the characteristic
equation

$\displaystyle\begin{array}[]{l}\left[\tilde{\bm{\beta}}_{0}^{(5)}\right]^{3}+\left(\frac{5}{k^{2}}-\frac{17}{2}\right)\left[\tilde{\bm{\beta}}_{0}^{(5)}\right]^{2}+\left(\frac{43k^{4}-48k^{2}+8}{2k^{4}}\right)\tilde{\bm{\beta}}_{0}^{(5)}+2\left(1-\frac{1}{k^{2}}\right)\left(\frac{4}{k^{2}}-7\right)=0,\end{array}$
(C.40)
$\displaystyle\begin{array}[]{l}\tilde{\bm{\beta}}_{0}^{(5)}=\frac{65}{16}-\frac{{\cal
E}}{4k^{2}}.\end{array}$

The Arscott condition implies three real and distinct values for
$\tilde{\bm{\beta}}_{0}^{(5)}$. We find that, $\mbox{if }{\cal
E}=16+k^{2}/4\mbox{ then }\tilde{\bm{\beta}}_{0}^{(5)}=4(k^{2}-1)/{k^{2}}$ and
Eq. (C.40) is fulfilled. The eigenfunction takes the form

$\displaystyle\begin{array}[]{l}\displaystyle\tilde{\bm{\Psi}}^{(5)}_{\;1}(u)=\frac{\bar{D}_{1}}{\mathrm{dn}^{1/2}u}\left(1-8\;\mathrm{sn}^{2}u\;\mathrm{cn}^{2}u\right),\qquad{\cal
E}=16+\frac{k^{2}}{4}\end{array}$ (C.43)

and in conjunction with $\tilde{\bm{\Psi}}^{(8)}_{\;2}(u)$ constitute a pair
of degenerate eigenfunctions as stated in [19]. In fact, the solution
mentioned in [19] comes from expansions in series of $\mathrm{sn}^{2}u$,
namely,

$\displaystyle\begin{array}[]{l}\displaystyle\tilde{\psi}^{(5)}(u)\stackrel{{\scriptstyle\text{(\ref{ps5-B}})}}{{=}}\frac{\tilde{{b}}_{\;0}^{(5)}}{\mathrm{dn}^{1/2}u}\left\\{1-\frac{1}{2}\left({\cal
E}-\frac{k^{2}}{4}\right)\mathrm{sn}^{2}u+\right.\vspace{3mm}\\\ \hskip
62.59596pt\left.\frac{k^{2}}{3}\left[4+\frac{1}{8k^{2}}\left({\cal
E}-\frac{k^{2}}{4}\right)\left({\cal
E}-4-\frac{9}{4}k^{2}\right)\right]\mathrm{sn}^{4}u\right\\},\end{array}$

which satisfy the characteristic equation

$\displaystyle\begin{array}[]{l}\lambda^{3}-4\left(\frac{7}{2}k^{2}+5\right)\lambda^{2}+16\left(\frac{3}{2}k^{4}+16k^{2}+4\right)\lambda-128k^{2}\left(3{k^{2}}+4\right)=0,\qquad\lambda={\cal
E}-\frac{k^{2}}{4}.\end{array}$

These solutions do not obey the Arscott condition, but we can see that
$\tilde{\psi}^{(5)}(u)=\tilde{\bm{\Psi}}^{(5)}(u)$ if ${\cal E}=16+k^{2}/4$.

##  Appendix D Svartholm’s and Erdelyi’s solutions

Svartholm [13] and Erdélyi [14, 15] have supposed that the coefficients of
their expansions in series of hypergeometric functions satisfy recurrence
relations having the form given in (72). Such conjecture has been repeated in
subsequent literature [3, 9, 21, 23], including Erdélyi himself [23]. However,
there are three forms for the recurrence relations, namely,

$\displaystyle\begin{array}[]{l}\alpha_{0}b_{1}+\beta_{0}b_{0}=0,\qquad\alpha_{n}b_{n+1}+\beta_{n}b_{n}+\gamma_{n}b_{n-1}=0,\quad(n\geq
1),\end{array}\vspace{3mm}$ (D.2)
$\displaystyle\begin{array}[]{r}\begin{cases}\alpha_{0}b_{1}+\beta_{0}b_{0}=0,\quad\alpha_{1}b_{2}+\beta_{1}b_{1}+\left[\gamma_{1}+\alpha_{-1}\right]b_{0}=0,\quad\vspace{2mm}\\\
\alpha_{n}b_{n+1}+\beta_{n}b_{n}+\gamma_{n}b_{n-1}=0,\quad(n\geq
2),\end{cases}\end{array}\vspace{3mm}$ (D.4)
$\displaystyle\begin{array}[]{l}\alpha_{0}b_{1}+\left[\beta_{0}+\alpha_{-1}\right]b_{0}=0,\qquad\alpha_{n}b_{n+1}+\beta_{n}b_{n}+\gamma_{n}b_{n-1}=0,\qquad(n\geq
1)\end{array}$ (D.6)

which reduce to (72) only if $\alpha_{-1}=0$.

The above recurrences also occur for some expansions in series of
hypergeometric functions and in series of Coulomb wave functions for solutions
of confluent and double-confluent Heun equations [46]. In fact, the procedure
used in Appendix A of reference [46] can be easily adapted for the expansions
of Svartholm and Erdélyi.

As an example we consider the Erdélyi solutions [15]

$\displaystyle\begin{array}[]{l}H(x)=\displaystyle\sum_{n=0}^{\infty}b_{n}P_{n}(x)=\displaystyle\sum_{n=0}^{\infty}b_{n}F(n+\lambda,-n+\mu;\gamma;x),\end{array}\vspace{2mm}$
(D.8) $\displaystyle\left[\gamma\neq 0,-1,-2,\cdots,\quad\mu-\lambda\neq
1,2,3,\cdots\right]$

where $\lambda$ and $\mu$ depend on the parameters of the Heun equation and
are connected by $\lambda+\mu=\gamma+\delta-1$. The condition $\mu-\lambda\neq
1,2,3,\cdots$ avoids two identical hypergeometric functions for different
values of $n$. For $H(x)$ given in (D.8), Eq. (2) leads to

$\displaystyle\displaystyle\sum_{n=0}^{\infty}{b}_{n}\left[\alpha_{n-1}\
P_{n-1}(x)+\beta_{n}\ P_{n}(x)+\gamma_{n+1}\ P_{n+1}(x)\right]=0,$ (D.9)

which can be written in Erdélyi’s notation [15] by taking $x=z$,
$b_{n}=c_{n}$, $\alpha_{n}=M_{n}$, $\beta_{n}=L_{n}$ and $\gamma_{n}=K_{n}$:
for example [3]

$\displaystyle\begin{array}[]{l}\alpha_{n}=M_{n}=-\frac{(n-\alpha+\lambda+1)(n-\beta+\lambda+1)(n-\gamma+\lambda+1)(n-\mu+1)}{\left(2n+\lambda-\mu+1\right)\left(2n+\lambda-\mu+2\right)}.\end{array}$

In order to get the three forms for the recurrence relations, we rewrite Eq.
(D.9) as

$\displaystyle\alpha_{-1}\ b_{0}\ P_{-1}(x)+\big{[}\alpha_{0}\
b_{1}+\beta_{0}\ b_{0}\big{]}P_{0}(x)+\big{[}\alpha_{1}\ b_{2}+\beta_{1}\
b_{1}+\gamma_{1}\ b_{0}\big{]}P_{1}(x)$
$\displaystyle+\displaystyle\sum_{n=2}^{\infty}\big{[}\alpha_{n}\
b_{n+1}+\beta_{n}\ b_{n}+\gamma_{n}\ b_{n-1}\big{]}P_{n}(x)=0.$ (D.11)

After this, we specify $\lambda$ and $\mu$ and equate to zero the coefficients
of each independent function $P_{n}(x)$. Next we regard only two choices for
($\lambda,\mu$).

In the first place, for the expansion mentioned at the end of Section 6 we
take $\lambda=\alpha$ and $\mu=\gamma+\delta-1-\alpha$. In this manner we
obtain the expansion $H^{(1)}(x)$,

$\displaystyle\begin{array}[]{l}H^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}b_{n}^{(1)}F(n+\alpha,-n-\alpha-1+\gamma+\delta;\gamma;x),\end{array}\quad$
(D.13) $\displaystyle\begin{array}[]{l}\left[\gamma\mbox{ and
}2+2\alpha-\gamma-\delta\neq 0,-1,-2,\cdots\right]\end{array},$ (D.15)

for which the recurrence relations for $b_{n}^{(1)}$ are given by

$\displaystyle\mbox{Eqs.}\;(\ref{r1})\;\mbox{ if }\;2\alpha-\gamma-\delta\neq
0,\ -1;\qquad\mbox{Eqs.}\;(\ref{r2})\mbox{ if
}2\alpha-\gamma-\delta=-1;\vspace{2mm}$
$\displaystyle\mbox{Eqs.}\;(\ref{r3})\;\mbox{ if }\;2\alpha-\gamma-\delta=0,$
(D.16)

with $\alpha_{n}$, $\beta_{n}$ and $\gamma_{n}$ defined as [15]

$\displaystyle\begin{cases}\begin{array}[]{l}\alpha_{n}^{(1)}=-\frac{(n+1)(n+1+\alpha-\beta)(n+1+\alpha-\gamma)(n+2+\alpha-\gamma-\delta)}{\left(2n+2+2\alpha-\gamma-\delta\right)\left(2n+3+2\alpha-\gamma-\delta\right)},\hskip
99.58464pt[n\geq-1]\vspace{3mm}\\\
\beta_{n}^{(1)}=\left(\frac{1}{2}-a\right)\Big{[}n(n+2\alpha-\gamma-\delta+1)+\alpha(\alpha+1-\gamma-\delta)\Big{]}-q+\frac{1}{2}\alpha\beta+\vspace{2mm}\\\
\hskip
28.45274pt\frac{1}{8}(\gamma-\delta)\left[2\alpha+2\beta-\gamma-\delta+\frac{(\gamma+\delta-2)(2\alpha-\gamma-\delta)(2\beta-\gamma-\delta)}{(2n+2\alpha-\gamma-\delta)(2n+2+2\alpha-\gamma-\delta)}\right],\quad[n\geq
0]\vspace{3mm}\\\
\gamma_{n}^{(1)}=-\frac{(n+2\alpha-\gamma-\delta)(n+\alpha-\delta)(n+\alpha-1)(n+\alpha+\beta-\gamma-\delta)}{(2n+2\alpha-\gamma-\delta)(2n+2\alpha-\gamma-\delta-1)},\hskip
96.73918pt[n\geq 1]\end{array}\end{cases}$ (D.17)

As to the recurrence relations which are missing in Erdély, they result
because there are terms linearly dependent in Eq. (D), namely,

$\displaystyle\begin{array}[]{l}\mbox{if }2\alpha-\gamma-\delta=-1,\quad
P_{-1}(x)=P_{1}(x)=F(\alpha-1,\alpha+1;\gamma;x)\Rightarrow\mbox{Eqs.}\;(\ref{r2});\end{array}\vspace{2mm}$
$\displaystyle\begin{array}[]{l}\mbox{if }2\alpha-\gamma-\delta=0,\qquad
P_{-1}(x)=P_{0}(x)=F(\alpha-1;\alpha;\gamma;x)\Rightarrow\mbox{Eqs.}\;(\ref{r3}).\end{array}$

Relations (D) take the place of (5.2), (5.3) and (5.4) given in Erdélyi’s
paper [15].

In the second place, the Svartholm solution arises when we put
$\lambda=\gamma+\delta-1$ and $\mu=0$ in (D.8). Then the Erdélyi expansion
(D.8) becomes

$\displaystyle\begin{array}[]{l}\tilde{H}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\tilde{b}_{n}^{(1)}\
F\left(n+\gamma+\delta-1,-n;\gamma;x\right),\end{array}\vspace{2mm}$ (D.21)
$\displaystyle\hskip 56.9055pt\begin{array}[]{l}\left[\gamma\neq
0,-1,-2,\cdots;\quad\gamma+\delta\neq 0,-1,-2,\cdots\right]\end{array}$

where the coefficients $\tilde{b}_{n}^{(1)}$ satisfy the recursions

$\displaystyle\mbox{Eqs.}\;(\ref{r1})\;\mbox{ if }\;\gamma+\delta\neq
1,2;\qquad\mbox{Eqs.}\;(\ref{r2})\mbox{ if }\gamma+\delta=1;\vspace{2mm}$
$\displaystyle\mbox{Eqs.}\;(\ref{r3})\;\mbox{ if }\;\gamma+\delta=2,$ (D.23)

with

$\displaystyle\begin{cases}\begin{array}[]{l}\tilde{\alpha}_{n}^{(1)}=-\frac{\left(n+1\right)\left(n+\gamma+\delta-\alpha\right)\left(n+\gamma+\delta-\beta\right)\left(n+\delta\right)}{\left(2n+\gamma+\delta\right)\left(2n+\gamma+\delta+1\right)},\hskip
128.0374pt[n\geq-1]\vspace{2mm}\\\
\tilde{\beta}_{n}^{(1)}=\left[\frac{1}{2}-a\right]n\left[n+\gamma+\delta-1\right]-q+\frac{1}{2}\alpha\beta+\vspace{2mm}\\\
\hskip
31.2982pt\frac{1}{8}(\gamma-\delta)\Big{[}2\alpha+2\beta-\gamma-\delta+\frac{(\gamma+\delta-2)(2\alpha-\gamma-\delta)(2\beta-\gamma-\delta)}{(2n+\gamma+\delta-2)(2n+\gamma+\delta)}\Big{]},\quad[n\geq
0]\vspace{2mm}\\\
\tilde{\gamma}_{n}^{(1)}=-\frac{\left(n+\alpha-1\right)\left(n+\beta-1\right)\left(n+\gamma+\delta-2\right)\left(n+\gamma-1\right)}{\left(2n+\gamma+\delta-3\right)\left(2n+\gamma+\delta-2\right)},\hskip
119.50148pt[n\geq 1]\end{array}\end{cases}$ (D.24)

By the homotopic transformations we can generate eight Svartholm-type
solutions, $\tilde{H}^{(i)}(x)$. These give series of trigonometric functions
for the associated Lamé equation. For this, we consider the definitions of the
basic elliptic functions [16]:

$\displaystyle\begin{array}[]{l}\operatorname{sn}u=\sin{v},\;\;\operatorname{cn}u=\cos{v},\;\;\operatorname{dn}u=\sqrt{1-k^{2}\sin^{2}v}\;\mbox{for}\;{u}=\int_{0}^{v}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}\theta}}.\end{array}$
(D.26)

In addition, for the associated Lamé equation
($x=\operatorname{sn}^{2}u=\sin^{2}v$), the relations [3]

$\displaystyle\begin{array}[]{l}F\left(-a,a;\frac{1}{2};\sin^{2}v\right)=\cos(2av),\qquad
F\left(a,1-a;\frac{3}{2};\sin^{2}v\right)=\frac{\cos[(2a-1)v]}{\cos{v}},\end{array}\vspace{2mm}$
(D.28)
$\displaystyle\begin{array}[]{l}F\left(1-a,a;\frac{3}{2};\sin^{2}v\right)=\frac{\sin[(2a-1)v]}{(2a-1)\sin{v}},\quad
F\left(a,2-a;\frac{3}{2};\sin^{2}v\right)=\frac{\sin[(2a-2)v]}{(a-1)\sin(2v)},\end{array}$
(D.30)

allow to get the trigonometric series. For example, for
$\gamma=\delta={1}/{2}$ we find

$\displaystyle\begin{array}[]{l}\tilde{H}^{(1)}(x)=\displaystyle\sum_{n=0}^{\infty}\begin{array}[]{l}\tilde{b}_{n}^{(1)}F\left(-n,n;\frac{1}{2};x\right)=\displaystyle\sum_{n=0}^{\infty}\tilde{b}_{n}^{(1)}\cos{(2nv)},\quad
x=\operatorname{sn}^{2}u=\sin^{2}{v},\end{array}\end{array}$ (D.33)

where the coefficients $\tilde{b}_{n}^{(1)}$ satisfy the relations (D.4) with

$\displaystyle\begin{cases}\begin{array}[]{l}\tilde{\alpha}_{n}^{(1)}=-\frac{1}{4}\left(n+1-\alpha\right)\left(n+1-\beta\right),\quad[n\geq-1],\vspace{2mm}\\\
\tilde{\beta}_{n}^{(1)}=\left(\frac{1}{2}-a\right)n^{2}-q+\frac{1}{2}\alpha\beta,\qquad[n\geq
0]\vspace{2mm}\\\
\tilde{\gamma}_{n}^{(1)}=-\frac{1}{4}{\left(n+\alpha-1\right)\left(n+\beta-1\right)},\qquad[n\geq
1]\end{array}\end{cases}$ (D.34)

where $a=1/k^{2}$. For the Lamé equation ($\gamma=\delta=\epsilon={1}/{2}$),
the expansions in trigonometric functions are equivalent to the solutions
found by Ince in 1940 [40].

Finally we find that, by performing the limits (124) for the confluent Heun
equation (123), the solutions (D.13)-(D) and (D.21)-(D) remain formally
unaltered while the coefficients (D.9) and (D.24) give, respectively,

$\displaystyle\begin{cases}\begin{array}[]{l}\alpha_{n}^{(1)}=-\rho\;\frac{(n+1)(n+1+\alpha-\gamma)(n+2+\alpha-\gamma-\delta)}{\left(2n+2+2\alpha-\gamma-\delta\right)\left(2n+3+2\alpha-\gamma-\delta\right)},\hskip
99.58464pt[n\geq-1],\vspace{3mm}\\\
\beta_{n}^{(1)}=-n(n+2\alpha-\gamma-\delta+1)-\alpha(\alpha+1-\gamma-\delta)+\sigma-\frac{1}{2}\alpha\rho+\vspace{2mm}\\\
\hskip
36.98866pt\frac{1}{4}\rho(\gamma-\delta)\left[\frac{(\gamma+\delta-2)(\gamma+\delta-2\alpha)}{(2n+2\alpha-\gamma-\delta)(2n+2+2\alpha-\gamma-\delta)}-1\right],\hskip
48.36958pt[n\geq 0],\vspace{3mm}\\\
\gamma_{n}^{(1)}=\rho\;\frac{(n+2\alpha-\gamma-\delta)(n+\alpha-\delta)(n+\alpha-1)}{(2n+2\alpha-\gamma-\delta)(2n+2\alpha-\gamma-\delta-1)},\hskip
128.0374pt[n\geq 1].\end{array}\end{cases}$ (D.35)
$\displaystyle\begin{cases}\begin{array}[]{l}\tilde{\alpha}_{n}^{(1)}=-\rho\frac{\left(n+1\right)\left(n+\gamma+\delta-\alpha\right)\left(n+\delta\right)}{\left(2n+\gamma+\delta\right)\left(2n+\gamma+\delta+1\right)},\hskip
128.0374pt[n\geq-1]\vspace{2mm}\\\
\tilde{\beta}_{n}^{(1)}=-n\left[n+\gamma+\delta-1\right]+\sigma-\frac{1}{2}\alpha\rho-\vspace{2mm}\\\
\hskip
31.2982pt\frac{1}{4}\rho(\gamma-\delta)\Big{[}1+\frac{(\gamma+\delta-2)(2\alpha-\gamma-\delta)}{(2n+\gamma+\delta-2)(2n+\gamma+\delta)}\Big{]},\quad[n\geq
0]\vspace{2mm}\\\
\tilde{\gamma}_{n}^{(1)}=\rho\frac{\left(n+\alpha-1\right)\left(n+\gamma+\delta-2\right)\left(n+\gamma-1\right)}{\left(2n+\gamma+\delta-3\right)\left(2n+\gamma+\delta-2\right)},\hskip
119.50148pt[n\geq 1]\end{array}\end{cases}$ (D.36)

Note that the three forms for the recurrence relations for the CHE have
already been obtained [46] independently of the solutions for the Heun
equation, as aforementioned.

## References

  * [1] Darboux, G.: Sur une équation linéaire. C. R. Acad. Sci. 94, 1645-1648 (1882).
  * [2] Humbert, P.: Fonctions de Lamé et fonctions de Mathieu. Mémorial des Sciences Mathématique, Vol. X, Gauthier-Villards (1926).
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