Eigenvalue Problems for Lamé’s Differential Equation

Eigenvalue Problems for Lamé’s Differential Equation

Hans VOLKMER

H. Volkmer

Department of Mathematical Sciences, University of Wisconsin-Milwaukee,
P.O. Box 413, Milwaukee, WI, 53201, USA<EMAIL_ADDRESS>
Received August 14, 2018, in final form December 06, 2018; Published online
December 12, 2018

The Floquet eigenvalue problem and a generalized form of the Wangerin
eigenvalue problem for Lamé’s differential equation are discussed. Results
include comparison theorems for eigenvalues and analytic continuation, zeros
and limiting cases of (generalized) Lamé–Wangerin eigenfunctions. Algebraic
Lamé functions and Lamé polynomials appear as special cases of Lamé–Wangerin
functions.

Lamé functions; singular Sturm–Liouville problems; tridiagonal matrices

33E10; 34B30

##  1 Introduction

The Lamé equation (Arscott [1, Chapter IX]) is

$\displaystyle\frac{{\rm d}^{2}w}{{\rm
d}z^{2}}+\big{(}h-\nu(\nu+1)k^{2}\operatorname{sn}^{2}(z,k)\big{)}w=0,$ (1.1)

where $\operatorname{sn}(z,k)$ is the Jacobian elliptic function with modulus
$k\in(0,1)$ (Whittaker and Watson [22, Chapter XXII]), $\nu\in\mathbb{R}$ and
$h$ is the eigenvalue parameter. This equation has regular singularities at
the points $z=2mK+{\rm i}(2n+1)K^{\prime}$, where $m$, $n$ are integers and
$K=K(k)$ and $K^{\prime}=K^{\prime}(k)$ denote complete elliptic integrals.
Various eigenvalue problems for the Lamé equation have been treated in the
literature.

The Lamé equation is an even Hill’s equation with fundamental period $2K$. The
theory of Hill’s equation is well-known; see, e.g., Arscott [1], Eastham [3]
and Magnus and Winkler [14]. Results on the periodic eigenvalue problem
specific for the Lamé equation with eigenfunctions satisfying $w(z+2K)=\pm
w(z)$ can be found in [6, Section 15,5], [9, 10] and [18, Chapter 29]. These
functions have many applications; see, e.g., [2]. In Section 2 of this paper
we will consider the more general Floquet eigenvalue problem $w(z+2K)={\rm
e}^{{\rm i}\mu\pi}w(z)$. For the general Hill’s equation this eigenvalue
problem is treated in Eastham [3]. Some results on the Floquet eigenvalue
problem specific for the Lamé equation can be found in Ince [8, Sections 7 and
8].

Wangerin [21] showed that Lamé’s equation appears when Laplace’s equation is
separated in confocal cyclidic coordinates of revolution. Such coordinate
systems can be found in Moon and Spencer [17] and in Miller [16]. They include
flat-ring, flat-disk, bi-cyclide and cap-cyclide coordinates. An outline of
Wangerin’s results is given in [6, Section 15.1.3]. In order to obtain
harmonic functions relevant for applications special solutions of the Lamé
equation called Lamé–Wangerin functions were introduced; see Erdélyi [5], [6,
p. 88]. The Lamé–Wangerin eigenvalue problem is obtained when we require that
$(\operatorname{sn}z)^{1/2}w(z)$ stays bounded at the singularities ${\rm
i}K^{\prime}$ and $2K+{\rm i}K^{\prime}$; see Erdélyi [5] and Erdélyi, Magnus
and Oberhettinger [6, Section 15.6]. These eigenfunctions will be defined on
the segment $({\rm i}K^{\prime},2K+{\rm i}K^{\prime})$ but can then be
continued analytically.

In Section 3 of this paper we consider a more general eigenvalue problem whose
eigenfunctions $w(z)$ have the form

$\displaystyle w(z)=(z-{\rm
i}K^{\prime})^{\nu+1}\sum_{n=0}^{\infty}q_{n}(z-{\rm i}K^{\prime})^{2n}$

at $z={\rm i}K^{\prime}$ and a similar condition at $z=2K+{\rm i}K^{\prime}$.
We call these eigenfunctions generalized Lamé–Wangerin functions. Every
classical Lamé–Wangerin function is also a generalized Lamé–Wangerin function
but not vice versa unless $\nu\geq-\frac{1}{2}$.

The motivation for introducing these functions is as follows. In Section 2 we
show that the eigenvalues of the Floquet eigenvalue problem agree with the
eigenvalues of an infinite tridiagonal matrix $F$ (considered in the Hilbert
sequence space $\ell^{2}(\mathbb{Z})$, $\mathbb{Z}$ the set of integers). One
is especially interested in the case that a matrix entry in the diagonal above
or below the main diagonal of $F$ vanishes because then the eigenvalue problem
splits in two problems whose eigenvalues are given by infinite tridiagonal
submatrices of $F$ that are only infinite in one direction (the underlying
Hilbert space can be taken as $\ell^{2}(\mathbb{N}_{0})$, $\mathbb{N}_{0}$ the
set of non-negative integers). It turns out that this special case occurs if
and only if $\nu+\mu$ or $\nu-\mu$ is an integer ($\nu$ is the parameter in
Lamé’s equation and $\mu$ is the parameter in Floquet’s condition.)
Interestingly, if $\nu+\mu$ or $\nu-\mu$ is an integer then the eigenvalues of
one of the submatrices are identical with the eigenvalues of a classical
Lamé–Wangerin problem. This is a simple observation but as far as I know has
not been stated in the literature. Of course, the obvious question is: If the
eigenvalues of one submatrix of $F$ are those for a classical Lamé–Wangerin
problem what is the meaning of the eigenvalues of the complementary submatrix?
As we show in this paper, these are the eigenvalues for a generalized
Lamé–Wangerin problem (non-classical except when $\nu=-\frac{1}{2}$).

A second motivation for introducing the generalized Lamé–Wangerin functions is
as follows. Lamé polynomials and algebraic Lamé functions are not special
cases of classical Lamé–Wangerin functions. However, they are special case of
generalized Lamé–Wangerin function. We show in Sections 6 and 7 how Lamé
polynomials and algebraic Lamé functions appear in the notation of generalized
Lamé–Wangerin functions. We should mention that we adopt the name “algebraic
Lamé functions” from [6, p. 68]. These functions are called “Lamé–Wangerin
functions” in Lambe [13] and non-meromorphic Lamé functions in Finkel et al.
[7].

In Section 5 we compare the eigenvalues of the Floquet and the Lamé–Wangerin
problems. In Sections 6 and 7 we show that algebraic Lamé functions and Lamé
polynomials are special cases of (generalized) Lamé–Wangerin functions. In
Section 8 we investigate the number of zeros of Lamé–Wangerin eigenfunctions.
In Section 9 we find the limit of Lamé–Wangerin functions as $k\to 0$.

I consider some of the results in this paper as new but not all results are
new. The treatment of the generalized Lamé–Wangerin problem is new. The
recursions (3.9) and (3.17) are known from [6] but the “symmetric” recursions
(3.12), (3.19) appear to be new. The latter recursions are used in some proofs
and also in Section 6. The results in Sections 4, 5, 8 and 9 are new. Lamé
polynomials and algebraic Lamé functions are well-known, so I make no claim
that Sections 6 and 7 contain new results.

##  2 Floquet solutions

On the real axis $z\in\mathbb{R}$, (1.1) is a Hill equation with fundamental
period $2K$. Let $\mu\in\mathbb{R}$. We call $h$ a Floquet eigenvalue if there
exists a nontrivial solution $w$ of (1.1) satisfying

$\displaystyle w(z+2K)={\rm e}^{{\rm i}\pi\mu}w(z),\qquad z\in\mathbb{R}.$
(2.1)

$w(z)$ is a corresponding Floquet eigenfunction. It is known [3, p. 31] that
the eigenvalues are real and form a sequence converging to $\infty$. We denote
the eigenvalues by

$\displaystyle h_{0}(\mu,\nu,k)\leq h_{1}(\mu,\nu,k)\leq
h_{2}(\mu,\nu,k)\leq\cdots.$

The eigenvalues are counted according to multiplicity. If $\mu$ is not an
integer then

$\displaystyle h_{0}(\mu,\nu,k)<h_{1}(\mu,\nu,k)<h_{2}(\mu,\nu,k)<\cdots.$

Obviously, we have

$\displaystyle
h_{m}(\mu,\nu,k)=h_{m}(\mu+2,\nu,k)=h_{m}(-\mu,\nu,k)=h_{m}(\mu,-\nu-1,k).$
(2.2)

Let $w_{1}(z,h,\nu,k)$ and $w_{2}(z,h,\nu,k)$ be the solutions of (1.1)
satisfying the initial conditions $w_{1}(0)=1$, $\frac{{\rm d}w_{1}}{{\rm
d}z}(0)=0$, $w_{2}(0)=0$, $\frac{{\rm d}w_{2}}{{\rm d}z}(0)=1$. Then Hill’s
discriminant $D$ is given by

$\displaystyle D(h,\nu,k)=w_{1}(2K,h,\nu,k)+\frac{{\rm d}w_{2}}{{\rm
d}z}(2K,h,\nu,k).$

The eigenvalues $h_{m}(\mu,\nu,k)$ are the solutions of the equation

$\displaystyle D(h,\nu,k)=2\cos(\mu\pi);$ (2.3)

see [3, equation (2.4.4)]. From (2.3) we easily obtain the following result
that will be needed later.

######  Theorem 2.1.

For every $m\in\mathbb{N}_{0}=\\{0,1,2,\dots\\}$, the function
$(\mu,\nu,k)\mapsto h_{m}(\mu,\nu,k)$ is continuous on
$\mathbb{R}\times\mathbb{R}\times[0,1)$.

Theorem 2.1 can also be inferred from results on Sturm–Liouville theory [12].

Following [6, p. 65] we transform (1.1) by setting

$\displaystyle t=\tfrac{1}{2}\pi-\operatorname{am}(z,k),$ (2.4)

where $\operatorname{am}$ is Jacobi’s amplitude function. We note that (2.4)
establishes a conformal mapping between the strip $|\Im z|<K^{\prime}$ and the
$t$-plane cut along the rays $m\pi\pm{\rm i}sL$, $s\geq 1$, $m\in\mathbb{Z}$,
where

$\displaystyle
L:=\operatorname{arccosh}\frac{1}{k}=\frac{1}{2}\ln\frac{1+k^{\prime}}{1-k^{\prime}},\qquad
k^{\prime}=\sqrt{1-k^{2}}.$

Then

$\displaystyle\operatorname{sn}z=\cos t,\qquad\operatorname{cn}z=\sin t.$

We obtain

$\displaystyle\big{(}1-k^{2}\cos^{2}t\big{)}\frac{{\rm d}^{2}w}{{\rm
d}t^{2}}+k^{2}\cos t\sin t\frac{{\rm d}w}{{\rm
d}t}+\big{(}h-\nu(\nu+1)k^{2}\cos^{2}t\big{)}w=0.$ (2.5)

Since $\operatorname{am}(z+2K)=\operatorname{am}z+\pi$, condition (2.1)
becomes

$\displaystyle w(t+\pi)={\rm e}^{-{\rm i}\pi\mu}w(t),\qquad t\in\mathbb{R}.$

This condition is equivalent to ${\rm e}^{{\rm i}\mu t}w(t)$ being periodic
with period $\pi$. Therefore, using Fourier series, eigenfunctions have the
form

$\displaystyle w(t)=\sum_{n=-\infty}^{\infty}c_{n}{\rm e}^{-{\rm
i}(\mu+2n)t}.$ (2.6)

By substituting (2.6) in (2.5), we obtain the three-term recursion

$\displaystyle\rho_{n}c_{n-1}+(\sigma_{n}-h)c_{n}+\tau_{n+1}c_{n+1}=0,\qquad
n\in\mathbb{Z},$ (2.7)

where

$\displaystyle\rho_{n}=-\tfrac{1}{4}k^{2}(2n-1+\mu+\nu)(2n-2+\mu-\nu),$
$\displaystyle\sigma_{n}=\tfrac{1}{2}k^{2}\nu(\nu+1)+\big{(}1-\tfrac{1}{2}k^{2}\big{)}(2n+\mu)^{2},$
$\displaystyle\tau_{n}=-\tfrac{1}{4}k^{2}(2n+\mu+\nu)(2n-1+\mu-\nu).$

This recursion is similar to the one given in [8, equation (7.1)] which is
based on Fourier cosine series instead of the complex form of Fourier series
we used. The behavior of solutions $\\{c_{n}\\}_{n\in\mathbb{Z}}$ of (2.7) as
$n\to\infty$ is given by Perron’s rule [19]. If $k\in(0,1)$ we choose $n_{0}$
so large that $\rho_{n}\neq 0$ and $\tau_{n+1}\neq 0$ for $n\geq n_{0}$. Then
the solutions $\\{c_{n}\\}_{n>n_{0}}$ of equations (2.7) for $n\geq n_{0}$
form a two-dimensional vector space. There exists a recessive solution which
is uniquely determined up to a constant factor with the property

$\displaystyle\lim_{n\to\infty}\frac{c_{n+1}}{c_{n}}=\frac{1-k^{\prime}}{1+k^{\prime}}<1.$
(2.8)

Every solution which is linearly independent of this solution satisfies

$\displaystyle\lim_{n\to\infty}\frac{c_{n+1}}{c_{n}}=\frac{1+k^{\prime}}{1-k^{\prime}}>1.$

Similar results hold for $n\to-\infty$. We obtain the following theorem.

######  Theorem 2.2.

Let $\mu,\nu\in\mathbb{R}$ and $k\in(0,1)$. Then $h$ is one of the eigenvalues
$h_{m}(\mu,\nu,k)$ if and only if the recursion (2.7) has a nontrivial
solution $\\{c_{n}\\}_{n\in\mathbb{Z}}$ such that

  1. $a)$

either there is $n_{0}$ such that $c_{n}=0$ for $n\geq n_{0}$ or $\\{c_{n}\\}$
is recessive as $n\to\infty$; and

  2. $b)$

either there is $n_{0}$ such that $c_{n}=0$ for $n\leq n_{0}$ or $\\{c_{n}\\}$
is recessive as $n\to-\infty$.

The expansion (2.6) of a corresponding eigenfunction converges in the strip
$|\Im t|<L$.

Of course, a nontrivial solution $\\{c_{n}\\}$ of (2.7) can be zero for $n\geq
n_{0}$ or $n\leq n_{0}$ only when one of the numbers $\rho_{n}$ or $\tau_{n}$
vanishes. This happens if and only if at least one of the numbers $\mu\pm\nu$
is an integer. These interesting cases will be discussed in Sections 5, 6 and
7.

Alternatively, we may expand

$\displaystyle
w(t)=\big{(}1-k^{2}\cos^{2}t\big{)}^{1/2}\sum_{n=-\infty}^{\infty}d_{n}{\rm
e}^{-{\rm i}(\mu+2n)t}.$

Then we obtain the “adjoint” recursion

$\displaystyle\tau_{n}d_{n-1}+(\sigma_{n}-h)d_{n}+\rho_{n+1}d_{n+1}=0,\qquad
n\in\mathbb{Z}.$ (2.9)

Theorem 2.2 also holds with (2.9) in place of (2.7).

##  3 Generalized Lamé–Wangerin functions

A (classical) Lamé–Wangerin function $w(z)$ is a nontrivial solution of Lamé’s
equation (1.1) with the property that $(\operatorname{sn}z)^{1/2}w(z)$ stays
bounded on the segment between the regular singularities $K^{\prime}$ and
$2K+{\rm i}K^{\prime}$; see [6, Section 15.6]. Such solutions exist only for
specific values of $h$. If we substitute $z=u+{\rm i}K^{\prime}$ then we
obtain the singular Sturm–Liouville problem [23]

$\displaystyle\frac{{\rm d}^{2}w}{{\rm
d}u^{2}}+\big{(}h-\nu(\nu+1)\operatorname{sn}^{-2}(u,k)\big{)}w=0,\qquad
0<u<2K,$ (3.1)

with the boundary condition that $(\operatorname{sn}u)^{-1/2}w$ stays bounded
on $(0,2K)$.

The eigenvalue problem splits into two problems, one for functions that are
even with respect to $K+{\rm i}K^{\prime}$, that is,

$\displaystyle w(K+{\rm i}K^{\prime}+s)=w(K+{\rm
i}K^{\prime}-s)\qquad\text{for $-K<s<K$},$ (3.2)

and one for functions which are odd with respect to $K+{\rm i}K^{\prime}$,
that is,

$\displaystyle w(K+{\rm i}K^{\prime}+s)=-w(K+{\rm
i}K^{\prime}-s)\qquad\text{for $-K<s<K$}.$ (3.3)

Without loss of generality, one may assume that $\nu\geq-\frac{1}{2}$, and
since the exponents at ${\rm i}K^{\prime}$ and $2K+{\rm i}K^{\prime}$ are
$\\{\nu+1,-\nu\\}$, a Lamé–Wangerin function has the form

$\displaystyle w(z)=(z-{\rm
i}K^{\prime})^{\nu+1}\sum_{n=0}^{\infty}q_{n}(z-{\rm i}K^{\prime})^{2n}$ (3.4)

for $z$ close to ${\rm i}K^{\prime}$ with $q_{0}\neq 0$.

We generalize these eigenvalue problems as follows. Let $\nu\in\mathbb{R}$,
$0<k<1$. We call $h\in\mathbb{C}$ an eigenvalue of the first Lamé–Wangerin
problem if (1.1) admits a nontrivial solution $w$ on the interval $({\rm
i}K^{\prime},2K+{\rm i}K^{\prime})$ which close to $z={\rm i}K^{\prime}$ has
the form (3.4) and satisfies $w^{\prime}(K+{\rm i}K^{\prime})=0$. The latter
property is equivalent to (3.2). The eigenfunction $w$ will be called a
Lamé–Wangerin function of the first kind. Note that we consider this
eigenvalue problem for all real $\nu$ not just for $\nu\geq-\frac{1}{2}$. Also
note that the condition $q_{0}\neq 0$ is not required in (3.4) although
$q_{0}\neq 0$ will hold if $\nu+\frac{1}{2}$ is not a negative integer.

Similarly, we call $h$ an eigenvalue of the second Lamé–Wangerin problem if
(1.1) admits a nontrivial solution $w$ on the interval $({\rm
i}K^{\prime},2K+{\rm i}K^{\prime})$ which close to $z={\rm i}K^{\prime}$ has
the form (3.4) and satisfies $w(K+{\rm i}K^{\prime})=0$. The latter property
is equivalent to (3.3). The eigenfunction $w$ will be called a Lamé–Wangerin
function of the second kind.

If $\nu>-\frac{3}{2}$ our eigenvalue problems are included in singular
Sturm–Liouville theory (see also [15]) but this theory does not give us
results for $\nu\leq-\frac{3}{2}$. We will treat these eigenvalue problems by
a different method developed below.

We substitute

$\displaystyle\eta={\rm e}^{-2{\rm i}t}$ (3.5)

in (2.5). We obtain the Fuchsian equation

$\displaystyle k^{2}\eta(\eta-\eta_{1})(\eta-\eta_{2})\left[\frac{{\rm
d}^{2}w}{{\rm
d}\eta^{2}}+\frac{1}{2}\left(\frac{1}{\eta}+\frac{1}{\eta-\eta_{1}}+\frac{1}{\eta-\eta_{2}}\right)\frac{{\rm
d}w}{{\rm d}\eta}\right]$
$\displaystyle\qquad{}+\left(h-k^{2}\nu(\nu+1)\frac{(1+\eta)^{2}}{4\eta}\right)w=0,$
(3.6)

where

$\displaystyle\eta_{1}:=\frac{1-k^{\prime}}{1+k^{\prime}}\in(0,1),\qquad\eta_{2}:=\frac{1+k^{\prime}}{1-k^{\prime}}\in(1,\infty).$

The differential equation (3.6) has regular singularities at
$\eta=0,\eta_{1},\eta_{2},\infty$ with exponents
$\big{\\{}{-}\frac{1}{2}\nu,\frac{1}{2}(\nu+1)\big{\\}}$,
$\big{\\{}0,\frac{1}{2}\big{\\}}$, $\big{\\{}0,\frac{1}{2}\big{\\}}$,
$\big{\\{}{-}\frac{1}{2}\nu,\frac{1}{2}(\nu+1)\big{\\}}$, respectively. If we
combine (2.4) with (3.5) we obtain

$\displaystyle\eta={\rm e}^{-2{\rm
i}(\frac{1}{2}\pi-\operatorname{am}z)}=(\operatorname{sn}z+{\rm
i}\operatorname{cn}z)^{-2}=(\operatorname{sn}z-{\rm
i}\operatorname{cn}z)^{2}.$ (3.7)

Setting $z=u+{\rm i}K^{\prime}$ for $0<u<K$ this gives

$\displaystyle\eta=\frac{1-\operatorname{dn}u}{1+\operatorname{dn}u}.$

This establishes a bijective increasing map between $u\in(0,K)$ and
$\eta\in(0,\eta_{1})$. Taking into consideration the behavior of $\eta$ close
to $u=0$ and $u=K$ we see that a Lamé–Wangerin function of the first kind
expressed in the variable $\eta$ is a solution of (3.6) on $(0,\eta_{1})$
which close to $\eta=0$ is of the form

$\displaystyle w(\eta)=\eta^{(\nu+1)/2}\sum_{n=0}^{\infty}c_{n}\eta^{n},$
(3.8)

and which is analytic at $\eta=\eta_{1}$. This implies that the radius of
convergence of the power series in (3.8) is $\geq\eta_{2}$. For the
coefficients $c_{n}$ we find the recursion

$\displaystyle\big{(}\beta^{(1)}_{0}-h\big{)}c_{0}+\gamma_{1}c_{1}=0,$
$\displaystyle\alpha_{n}c_{n-1}+\big{(}\beta_{n}^{(1)}-h\big{)}c_{n}+\gamma_{n+1}c_{n+1}=0,\qquad
n\geq 1,$ (3.9)

where

$\displaystyle\alpha_{n}=-\tfrac{1}{2}k^{2}(n+\nu)(2n-1),$
$\displaystyle\beta_{n}^{(1)}=\tfrac{1}{2}k^{2}\nu(\nu+1)+\big{(}1-\tfrac{1}{2}k^{2}\big{)}(2n+\nu+1)^{2},$
$\displaystyle\gamma_{n}=-\tfrac{1}{2}k^{2}(2n+2\nu+1)n.$

Note that the equations (3.9) for $n\geq 1$ agree with (2.7) when we set
$\mu=\nu+1$. The recursion (3.9) is given in [6, Section 15.6(15)].

Using Perron’s rule, we see that $h$ is an eigenvalue of the first
Lamé–Wangerin problem if and only if (3.9) has a nontrivial solution
$\\{c_{n}\\}_{n=0}^{\infty}$ which is either identically zero for large $n$ or
satisfies (2.8). Of course, a nontrivial solution $\\{c_{n}\\}_{n=0}^{\infty}$
of (3.9) can be identically zero for large $n$ only if one of the numbers
$\alpha_{n}$ is zero, that is, if $\nu$ is a negative integer.

Alternatively, we may expand a Lamé–Wangerin function of the first kind in the
form

$\displaystyle
w(\eta)=\eta^{(\nu+1)/2}(\eta_{2}-\eta)^{1/2}\sum_{n=0}^{\infty}a_{n}\eta^{n}$
(3.10)

with the power series having radius $\geq\eta_{2}$. In order to find the
recursion for the coefficients $a_{n}$ we transform (3.6) by setting

$\displaystyle w(\eta)=(\eta_{2}-\eta)^{1/2}v(\eta)$

to

$\displaystyle k^{2}\eta(\eta-\eta_{1})(\eta-\eta_{2})\left[\frac{{\rm
d}^{2}v}{{\rm
d}\eta^{2}}+\frac{1}{2}\left(\frac{1}{\eta}+\frac{1}{\eta-\eta_{1}}+\frac{3}{\eta-\eta_{2}}\right)\frac{{\rm
d}v}{{\rm d}\eta}\right]$
$\displaystyle\qquad{}+\left(h-k^{2}\nu(\nu+1)\frac{(1+\eta)^{2}}{4\eta}+\tfrac{1}{4}k^{2}(2\eta-\eta_{1})\right)v=0.$
(3.11)

We obtain the recursion

$\displaystyle\big{(}\epsilon^{(1)}_{0}-h\big{)}a_{0}+\delta_{1}a_{1}=0,$
$\displaystyle\delta_{n}a_{n-1}+\big{(}\epsilon_{n}^{(1)}-h\big{)}a_{n}+\delta_{n+1}a_{n+1}=0,\qquad
n\geq 1,$ (3.12)

where

$\displaystyle\delta_{n}=-\tfrac{1}{2}k^{2}n(2n+2\nu+1),$
$\displaystyle\epsilon_{n}^{(1)}=\tfrac{1}{2}k^{2}\nu(\nu+1)+\big{(}1-\tfrac{1}{2}k^{2}\big{)}(2n+\nu+1)^{2}+\tfrac{1}{4}k^{2}\eta_{1}(4n+2\nu+3)$
$\displaystyle\hphantom{\epsilon_{n}^{(1)}}{}=\tfrac{1}{2}k^{2}\nu(\nu+1)-k^{\prime}\big{(}2n+\tfrac{3}{2}+\nu\big{)}+\big{(}1-\tfrac{1}{2}k^{2}\big{)}\big{(}\tfrac{1}{4}+\big{(}2n+\tfrac{3}{2}+\nu\big{)}^{2}\big{)}.$

It is a pleasant surprise that, in contrast to (3.9), recursion (3.12) is of
self-adjoint form. We take advantage of this observation and introduce a
symmetric operator $S=S^{(1)}(\nu,k)$ in the Hilbert space
$\ell^{2}(\mathbb{N}_{0})$ with the standard inner product. The domain of
definition of $S$ is

$\displaystyle
D(S)=\left\\{\\{x_{n}\\}_{n=0}^{\infty}\colon\sum_{n=0}^{\infty}n^{4}|x_{n}|^{2}<\infty\right\\}$

and $S$ is defined on $D(S)$ by

$\displaystyle S(\\{x_{j}\\})_{0}=\epsilon_{0}^{(1)}x_{0}+\delta_{1}x_{1},$
$\displaystyle
S(\\{x_{j}\\})_{n}=\delta_{n}x_{n-1}+\epsilon_{n}^{(1)}x_{n}+\delta_{n+1}x_{n+1},\qquad
n\geq 1.$

So $S$ is represented by an infinite symmetric tridiagonal matrix.

######  Theorem 3.1.

Let $\nu\in\mathbb{R}$ and $k\in[0,1)$.

  1. $(a)$

$S^{(1)}(\nu,k)$ is a self-adjoint operator in $\ell^{2}(\mathbb{N}_{0})$ with
compact resolvent and bounded below.

  2. $(b)$

If $k\in(0,1)$ the eigenvalues of $S^{(1)}(\nu,k)$ agree with the eigenvalues
of the first Lamé–Wangerin problem.

  3. $(c)$

If $k\in(0,1)$ the eigenvalues of $S^{(1)}(\nu,k)$ are simple.

###### Proof.

(a) We abbreviate $S=S^{(1)}(\nu,k)$, and write $S=A+B$ with
$A=S^{(1)}(\nu,0)$. So $A$ is represented by an infinite diagonal matrix with
diagonal entries $(2n+\nu+1)^{2}$, $n\in\mathbb{N}_{0}$. It is clear that $A$
is a positive semi-definite self-adjoint operator with compact resolvent.
There are two constants $\lambda>0$ and $c\in(0,1)$ such that

$\displaystyle\|Bx\|\leq c\|(A+\lambda)x\|\qquad\text{for all $x\in D(S)$}.$
(3.13)

To prove this it is convenient to write $B=B_{1}+B_{2}+B_{3}$ where each
$B_{i}$ has a matrix representation consisting of only one nonzero “diagonal”,
and estimate $\|Bx\|\leq\|B_{1}x\|+\|B_{2}x\|+\|B_{3}x\|$. We can reach $c<1$
because the factor of $n^{2}$ on the main diagonal of $A$ is $4$ while the
factors of $n^{2}$ on the three diagonals of $B$ are $-k^{2}$, $-2k^{2}$,
$-k^{2}$, respectively. From (3.13) we obtain that $T:=B(A+\lambda)^{-1}$ is a
bounded linear operator with operator norm $\|T\|\leq c<1$. Therefore, $1+T$
is invertible and

$\displaystyle(S+\lambda)^{-1}=(A+\lambda+B)^{-1}=(A+\lambda)^{-1}(1+T)^{-1}.$

This shows that $(S+\lambda)^{-1}$ is a compact operator. Since $S$ is
symmetric, we find that $S$ is self-adjoint; compare [11, Chapter V, Theorem
4.3]. From (3.13) we also obtain that $S+\lambda$ is positive definite [11,
Chapter V, Theorem 4.11]. Therefore, (a) follows.

(b) $h$ is an eigenvalue of $S$ if and only if the recursion (3.12) has a
nontrivial solution $\\{a_{n}\\}_{n=0}^{\infty}$ with the property that
$\sum\limits_{n=0}^{\infty}n^{4}|a_{n}|^{2}<\infty$. By Perron’s rule the
latter property is equivalent to $a_{n}=0$ for large $n$ or $\\{a_{n}\\}$ is
recessive as $n\to\infty$.

(c) If $k\in(0,1)$ the eigenvalues of $S$ are simple because the corresponding
eigenfunctions of the first Lamé–Wangerin problem are even with respect to
$K+{\rm i}K^{\prime}$. ∎

Based on Theorem 3.1 we write the eigenvalues of the first Lamé–Wangerin
problem with $k\in(0,1)$ in the form

$\displaystyle
H^{(1)}_{0}(\nu,k)<H^{(1)}_{1}(\nu,k)<H^{(1)}_{2}(\nu,k)<\cdots.$

The Lamé–Wangerin function belonging to $H^{(1)}_{m}(\nu,k)$ will be denoted
by $w_{m}^{(1)}(z,\nu,k)$. If a normalization is required it will be stated
separately. We note that the corresponding eigenvectors
$\\{a_{n}\\}_{n=0}^{\infty}$ of $S$ when properly normalized form an
orthonormal basis in the Hilbert space $\ell^{2}(\mathbb{N}_{0})$.

The eigenvalues of $S^{(1)}(\nu,0)$ are $(2n+\nu+1)^{2}$ for
$n\in\mathbb{N}_{0}$. If we arrange this sequence in increasing order repeated
according to multiplicity we denote these eigenvalues by $H^{(1)}_{m}(\nu,0)$.
Explicitly, they are given by the following lemma.

######  Lemma 3.2.

Let $p-1<\nu\leq p$ with $p\in\mathbb{Z}$. Then, for all $m\in\mathbb{N}_{0}$,

$\displaystyle H^{(1)}_{m}(\nu,0)=(2\ell+\nu+1)^{2},$

where

$\displaystyle\ell=\begin{cases}m&\text{if $m+p\geq 0$},\\\
\frac{1}{2}(m-p)&\text{if $m+p<0$, $m+p$ even},\\\
\frac{1}{2}(-m-p-1)&\text{if $m+p<0$, $m+p$ odd}.\\\ \end{cases}$

We will need continuity of the eigenvalues $H^{(1)}_{m}(\nu,k)$.

######  Theorem 3.3.

The function $(\nu,k)\mapsto H^{(1)}_{m}(\nu,k)$ is continuous on
$\mathbb{R}\times[0,1)$ for every $m\in\mathbb{N}_{0}$.

###### Proof.

Set $S(\nu,k)=S^{(1)}(\nu,k)$, $A(\nu)=S(\nu,0)$ and
$S(\nu,k)=A(\nu)+B(\nu,k)$. Let $\nu_{0}>0$ and $k_{0}\in(0,1)$ be given, and
set $\Omega:=[-\nu_{0},\nu_{0}]\times[0,k_{0}]$. Then we can find $\lambda>0$
large enough and $c\in(0,1)$ such that (3.13) holds uniformly for
$(\nu,k)\in\Omega$. It follows that $T(\nu,k):=B(\nu,k)(A(\nu)+\lambda)^{-1}$
is a bounded linear operator with operator norm $\|T(\nu,k)\|\leq c$ for all
$(\nu,k)\in\Omega$. As before, we have

$\displaystyle(S(\nu,k)+\lambda)^{-1}=(A(\nu)+\lambda)^{-1}(1+T(\nu,k))^{-1},\qquad(\nu,k)\in\Omega.$
(3.14)

Suppose we have a sequence $(\nu_{n},k_{n})\in\Omega$ which converges to
$\big{(}\hat{\nu},\hat{k}\big{)}$ as $n\to\infty$. Then we can easily show
using the definitions of $A$ and $T$ that

$\displaystyle\big{\|}(A(\nu_{n})+\lambda)^{-1}-(A(\hat{\nu})+\lambda)^{-1}\big{\|}\to
0\qquad\text{as $n\to\infty$},$

and

$\displaystyle\big{\|}T(\nu_{n},k_{n})-T\big{(}\hat{\nu},\hat{k}\big{)}\big{\|}\to
0\qquad\text{as $n\to\infty$}.$

Using (3.14) we then obtain that

$\displaystyle\big{\|}(S(\nu_{n},k_{n})+\lambda)^{-1}-\big{(}S\big{(}\hat{\nu},\hat{k}\big{)}+\lambda\big{)}^{-1}\big{\|}\to
0\qquad\text{as $n\to\infty$.}$ (3.15)

If $K_{n}$ is a sequence of positive definite compact Hermitian operators
converging to a positive definite compact Hermitian operator $K$ with respect
to the operator norm, then the $m$th largest eigenvalue of $K$ (counted
according to multiplicity) converges to the $m$ largest eigenvalue of $K$ as
$n\to\infty$ for every $m\in\mathbb{N}_{0}$. This follows directly from the
minimum-maximum-principle If we set $K_{n}=(S(\nu_{n},k_{n})+\lambda)^{-1}$,
$K=\big{(}S\big{(}\hat{\nu},\hat{k}\big{)}+\lambda\big{)}^{-1}$ and use (3.15)
we obtain $H^{(1)}_{m}(\nu_{n},k_{n})\to
H^{(1)}_{m}\big{(}\hat{\nu},\hat{k}\big{)}$ as $n\to\infty$ for every
$m\in\mathbb{N}_{0}$ as desired. ∎

A Lamé–Wangerin function of the second kind can be written in the form

$\displaystyle
w(\eta)=\eta^{(\nu+1)/2}(\eta_{1}-\eta)^{1/2}(\eta_{2}-\eta)^{1/2}\sum_{n=0}^{\infty}d_{n}\eta^{n},$
(3.16)

where the power series $\sum d_{n}\eta^{n}$ has radius $\geq\eta_{2}$. If we
set

$\displaystyle w(\eta)=(\eta-\eta_{1})^{1/2}(\eta-\eta_{2})^{1/2}v(\eta)$

in (3.6), we obtain

$\displaystyle k^{2}\eta(\eta-\eta_{1})(\eta-\eta_{2})\left[\frac{{\rm
d}^{2}v}{{\rm
d}\eta^{2}}+\frac{1}{2}\left(\frac{1}{\eta}+\frac{3}{\eta-\eta_{1}}+\frac{3}{\eta-\eta_{2}}\right)\frac{{\rm
d}v}{{\rm d}\eta}\right]$
$\displaystyle\qquad{}+\left(\tfrac{3}{2}k^{2}\eta-1+\tfrac{1}{2}k^{2}+h-k^{2}\nu(\nu+1)\frac{(1+\eta)^{2}}{4\eta}\right)v=0.$

This gives the recursion

$\displaystyle\big{(}\beta_{0}^{(2)}-h\big{)}d_{0}+\gamma_{1}d_{1}=0,$
$\displaystyle\alpha_{n+1}d_{n-1}+\big{(}\beta_{n}^{(2)}-h\big{)}d_{n}+\gamma_{n+1}d_{n+1}=0,\quad
n\geq 1,$ (3.17)

where $\alpha_{n}$, $\gamma_{n}$ are as in (3.9) and

$\displaystyle\beta_{n}^{(2)}=\tfrac{1}{2}k^{2}\nu(\nu+1)+\big{(}1-\tfrac{1}{2}k^{2}\big{)}(2n+\nu+2)^{2}.$

Note that the equations (3.17) for $n\geq 1$ agree with (2.9) when we set
$\mu=\nu+2$. The recursion (3.17) is given in [6, Section 15.6(16)].

Alternatively, a Lamé–Wangerin function of the second kind can be written as

$\displaystyle
w(\eta)=\eta^{(\nu+1)/2}(\eta_{1}-\eta)^{1/2}\sum_{n=0}^{\infty}b_{n}\eta^{n},$
(3.18)

where the power series $\sum b_{n}\eta^{n}$ has radius $\geq\eta_{2}$. If we
set

$\displaystyle w(\eta)=(\eta_{1}-\eta)^{1/2}v(\eta)$

in (3.6), we obtain (3.11) with $\eta_{1}$, $\eta_{2}$ interchanged. This
gives the recursion

$\displaystyle\big{(}\epsilon_{0}^{(2)}-h\big{)}b_{0}+\delta_{1}b_{1}=0,$
$\displaystyle\delta_{n}b_{n-1}+\big{(}\epsilon_{n}^{(2)}-h\big{)}b_{n}+\delta_{n+1}b_{n+1}=0,\qquad
n\geq 1,$ (3.19)

where

$\displaystyle\delta_{n}=-\tfrac{1}{2}k^{2}n(2n+2\nu+1),$
$\displaystyle\epsilon_{n}^{(2)}=\tfrac{1}{2}k^{2}\nu(\nu+1)+\big{(}1-\tfrac{1}{2}k^{2}\big{)}(2n+\nu+1)^{2}+\tfrac{1}{4}k^{2}\eta_{2}(4n+2\nu+3)$
$\displaystyle\hphantom{\epsilon_{n}^{(2)}}{}=\tfrac{1}{2}k^{2}\nu(\nu+1)+k^{\prime}\big{(}2n+\tfrac{3}{2}+\nu\big{)}+\big{(}1-\tfrac{1}{2}k^{2}\big{)}\big{(}\tfrac{1}{4}+\big{(}2n+\tfrac{3}{2}+\nu\big{)}^{2}\big{)}.$

For the second Lamé–Wangerin problem we have results parallel to Theorem 3.1,
Lemma 3.2 and Theorem 3.3. We denote the eigenvalues of the second
Lamé–Wangerin problem by $H^{(2)}_{m}(\nu,k)$, and the corresponding
Lamé–Wangerin eigenfunctions by $w^{(2)}_{m}(z,\nu,k)$.

##  4 Analytic continuation of Lamé–Wangerin functions

In the previous section Lamé–Wangerin functions were defined on the interval
$({\rm i}K^{\prime},2K+{\rm i}K^{\prime})$. We analytically continue these
functions to the strip $0\leq\Im z<K^{\prime}$ as follows. Using (3.7) and
(3.8) a Lamé–Wangerin function $w^{(1)}$ of the first kind can be written as

$\displaystyle w^{(1)}(z)={\rm e}^{-{\rm
i}(\nu+1)(\frac{1}{2}\pi-\operatorname{am}z)}\sum_{n=0}^{\infty}c_{n}(\operatorname{sn}z-{\rm
i}\operatorname{cn}z)^{2n}.$ (4.1)

Since the power series $\sum c_{n}\eta^{n}$ has radius larger than $1$ and
$|\eta|\leq 1$ for $0\leq\Im z<K^{\prime}$, the expansion (4.1) converges in
the strip $0\leq\Im z<K^{\prime}$.

If $0<\eta<\eta_{1}$ we have

$\displaystyle\frac{1}{4}k^{2}\big{(}\eta^{1/2}+\eta^{-1/2}\big{)}^{2}-1=\frac{1}{4}k^{2}\eta^{-1}(\eta_{1}-\eta)(\eta_{2}-\eta).$
(4.2)

If $z$ is on the segment $({\rm i}K^{\prime},K+{\rm i}K^{\prime})$ and $\eta$
is given by (3.7) then (4.2) implies

$\displaystyle{\rm
i}\operatorname{dn}z=\frac{1}{2}k\eta^{-1/2}(\eta_{1}-\eta)^{1/2}(\eta_{2}-\eta)^{1/2}.$
(4.3)

Therefore, (3.16) implies

$\displaystyle w^{(2)}(z)=2{\rm i}k^{-1}{\rm e}^{-{\rm
i}(\nu+2)(\frac{1}{2}\pi-\operatorname{am}z)}\operatorname{dn}z\sum_{n=0}^{\infty}d_{n}(\operatorname{sn}z-{\rm
i}\operatorname{cn}z)^{2n}.$ (4.4)

Again, this expansion is convergent in the strip $0\leq\Im z<K^{\prime}$.

In order to deal with expansions (3.10) and (3.18) we introduce the function

$\displaystyle I_{1}(z):=(\operatorname{dn}z+\operatorname{cn}z)^{1/2}$ (4.5)

also appearing in [8]. This function is analytic in the strip $-K^{\prime}<\Im
z<K^{\prime}$ when the branch of the root is chosen as follows. The function
$\operatorname{dn}z+\operatorname{cn}z$ does not assume negative values or
zero in the rectangle $-2K<\Re z<2K$, $-K^{\prime}<\Im z<K^{\prime}$. We
choose the principal branch of the root in (4.5) in this rectangle. We choose
positive imaginary roots on the segments $(-2K,-2K+{\rm i}K^{\prime})$ and
$(2K-{\rm i}K^{\prime},2K)$. For other $z$, $I_{1}(z)$ is determined by
$I_{1}(z+4K)=-I_{1}(z)$. A second function is defined by

$\displaystyle
I_{2}(z):=-I_{1}(z+2K)=-(\operatorname{dn}z-\operatorname{cn}z)^{1/2}.$

For $0<\eta<\eta_{1}$ we have the identity

$\displaystyle(1-\eta+q)^{1/2}+(1-\eta-q)^{1/2}=2^{1/2}(1-k^{\prime})^{1/2}(\eta_{2}-\eta)^{1/2},$
(4.6)

where

$\displaystyle q=k(\eta_{1}-\eta)^{1/2}(\eta_{2}-\eta)^{1/2}$

and all roots denote positive roots of positive numbers. For $z$ between ${\rm
i}K^{\prime}$ and $K+{\rm i}K^{\prime}$ we have

$\displaystyle{\rm
i}\operatorname{cn}z=\frac{1}{2}\big{(}\eta^{-1/2}-\eta^{1/2}\big{)}.$

Therefore, it follows from (4.3) and (4.6) that the analytic continuation of
$J_{1}=\eta^{-1/4}(\eta_{2}-\eta)^{1/2}$ to the strip $-K^{\prime}<\Im
z<K^{\prime}$ is given by

$\displaystyle(1-k^{\prime})^{1/2}J_{1}(z)={\rm e}^{{\rm
i}\frac{1}{4}\pi}I_{1}(z)+{\rm e}^{-{\rm i}\frac{1}{4}\pi}I_{2}(z).$ (4.7)

Therefore, the analytic continuation of the Lamé–Wangerin function $w^{(1)}$
of the first kind given by (3.10) to the strip $0\leq\Im z<K^{\prime}$ is

$\displaystyle w^{(1)}(z)={\rm e}^{-{\rm
i}(\nu+\frac{3}{2})(\frac{1}{2}\pi-\operatorname{am}z)}J_{1}(z)\sum_{n=0}^{\infty}a_{n}(\operatorname{sn}z-{\rm
i}\operatorname{cn}z)^{2n}.$

Similarly, for $0<\eta<\eta_{1}$ we have

$\displaystyle(1-\eta+q)^{1/2}-(1-\eta-q)^{1/2}=2^{1/2}(1+k^{\prime})^{1/2}(\eta_{1}-\eta)^{1/2}.$

It follows that the analytic continuation of the function
$J_{2}=\eta^{-1/4}(\eta_{1}-\eta)^{1/2}$ to the strip $-K^{\prime}<\Im
z<K^{\prime}$ is given by

$\displaystyle(1+k^{\prime})^{1/2}J_{2}(z)={\rm e}^{{\rm
i}\frac{1}{4}\pi}I_{1}(z)-{\rm e}^{-{\rm i}\frac{1}{4}\pi}I_{2}(z).$ (4.8)

If a Lamé–Wangerin function $w^{(2)}$ of the second kind is given by (3.18)
then its analytic continuation is

$\displaystyle w^{(2)}(z)={\rm e}^{-{\rm
i}(\nu+\frac{3}{2})(\frac{1}{2}\pi-\operatorname{am}z)}J_{2}(z)\sum_{n=0}^{\infty}b_{n}(\operatorname{sn}z-{\rm
i}\operatorname{cn}z)^{2n}.$

##  5 Comparison of eigenvalues

Every Lamé–Wangerin function is also a Floquet eigenfunction.

######  Lemma 5.1.

Let $\nu\in\mathbb{R}$ and $0<k<1$. A Lamé–Wangerin function $w^{(1)}(z)$ of
the first kind satisfies

$\displaystyle w^{(1)}(z+2K)={\rm e}^{{\rm i}(\nu+1)\pi}w^{(1)}(z).$

A Lamé–Wangerin function $w^{(2)}(z)$ of the second kind satisfies

$\displaystyle w^{(2)}(z+2K)={\rm e}^{{\rm i}\nu\pi}w^{(2)}(z).$

###### Proof.

This follows from (4.1) and (4.4). ∎

If $\mu+\nu$ or $\mu-\nu$ is an integer then the Floquet eigenvalues
$h_{m}(\mu,\nu)$ can be expressed in terms of Lamé–Wangerin eigenvalues
$H_{m}^{(j)}(\nu)$. The properties (2.2) show that it is sufficient to
consider the case $\mu=\nu+1$. Then we have the following result.

######  Theorem 5.2.

Let $0<k<1$, $\nu\in\mathbb{R}$, $p\in\mathbb{N}_{0}$ and $p-1<|\nu|\leq p$.

  1. $(a)$

If $\nu\geq 0$ then

$\displaystyle h_{m}(\nu+1,\nu)=H^{(2)}_{m}(-\nu-1),\qquad m=0,1,\dots,p-1,$
$\displaystyle h_{p+2i+1}(\nu+1,\nu)=H^{(2)}_{p+i}(-\nu-1),\qquad i\geq 0,$
$\displaystyle h_{p+2i}(\nu+1,\nu)=H^{(1)}_{i}(\nu),\qquad i\geq 0.$

  2. $(b)$

If $\nu<0$ then

$\displaystyle h_{m}(\nu+1,\nu)=H^{(1)}_{m}(\nu),\qquad m=0,1,\dots,p-1,$
$\displaystyle h_{p+2i+1}(\nu+1,\nu)=H^{(1)}_{p+i}(\nu),\qquad i\geq 0,$
$\displaystyle h_{p+2i}(\nu+1,\nu)=H^{(2)}_{i}(-\nu-1),\qquad i\geq 0.$

  3. $(c)$

If $\nu$ is an integer then

$\displaystyle h_{0}(\nu+1,\nu)<h_{1}(\nu+1,\nu)<\dots<h_{p}(\nu+1,\nu),$

and, for $i\geq 0$,

$\displaystyle
h_{p+2i}(\nu+1,\nu)=h_{p+2i+1}(\nu+1,\nu)<h_{p+2i+2}(\nu+1,\nu)=h_{p+2i+3}(\nu+1,\nu)<\cdots.$

###### Proof.

(a), (b) Suppose first that $\nu$ is not an integer. Then the eigenvalues
$f_{m}(k):=h_{m}(\nu+1,\nu,k)$ form a strictly increasing sequence. By Lemma
5.1, each eigenvalue $g_{m}(k):=H^{(1)}_{m}(\nu,k)$ and
$G_{m}(k):=H^{(2)}_{m}(-\nu-1,k)$ is among the $f$-eigenvalues. This is also
true for $k=0$. The sequence $\\{f_{m}(0)\\}_{m=0}^{\infty}$ is given by the
sequence $\big{\\{}(2n+\nu+1)^{2}\big{\\}}_{n\in\mathbb{Z}}$ when arranged in
increasing order, $\\{g_{m}(0)\\}$ is given by
$\\{(2n+\nu+1)^{2}\\}_{n=0}^{\infty}$ in increasing order and $\\{G_{m}(0)\\}$
is given by $\big{\\{}(2n+\nu+1)^{2}\big{\\}}_{n=-\infty}^{-1}$ in increasing
order. Because of continuity of the functions $f_{m}$, $g_{m}$, $G_{m}$
(Theorems 2.1 and 3.3) the order of these eigenvalues is the same for all
$k\in[0,1)$. An analysis of the order at $k=0$ proves (a) and (b) for
noninteger $\nu$. The result extends to integer $\nu$ by continuity.

(c) Let $\nu$ be an integer. Then we apply (a) or (b) to $\nu+\epsilon$ in
place of $\nu$ and take limits $\epsilon\to 0^{\pm}$. This proves (c). ∎

We now compare the eigenvalues $H_{m}^{(j)}(\nu)$ with $H_{m}^{(j)}(-\nu-1)$.

######  Theorem 5.3.

Let $p\in\mathbb{N}_{0}$, $0<k<1$. Let either $H=H^{(1)}$ or $H=H^{(2)}$.

  1. $(a)$

If $-p-\tfrac{3}{2}<\nu<-p-\tfrac{1}{2}$ then

$\displaystyle
H_{p}(\nu)<H_{0}(-\nu-1)<H_{p+1}(\nu)<H_{1}(-\nu-1)<H_{p+2}(\nu)<\cdots.$

  2. $(b)$

If $\nu=-p-\tfrac{1}{2}$ then

$\displaystyle
H_{p-1}(\nu)<H_{0}(-\nu-1)=H_{p}(\nu)<H_{1}(-\nu-1)=H_{p+1}(\nu)<\cdots.$

###### Proof.

We consider the eigenvalues $H_{m}=H^{(1)}_{m}$. The proof for
$H_{m}=H^{(2)}_{m}$ is similar.

(a) Let $-p-\tfrac{3}{2}<\nu<-p-\tfrac{1}{2}$. The eigenvalues $H_{m}(\nu)$,
$m\geq 0$, are pairwise distinct, and the eigenvalues $H_{\ell}(-\nu-1)$,
$\ell\geq 0$, are pairwise distinct. The eigenvalues $H_{m}(\nu)$ are also
distinct from $H_{\ell}(-\nu-1)$ because the corresponding eigenfunctions are
linearly independent. Therefore, using continuity of the functions $k\mapsto
H_{m}(\nu,k)$ (Theorem 3.3) the order of the eigenvalues $H_{m}(\nu,k)$,
$H_{\ell}(-\nu-1,k)$ must be the same for all $k\in[0,1)$. The sequence
$\\{H_{m}(\nu,0)\\}_{m=0}^{\infty}$ agrees with
$\big{\\{}(2n+\nu+1)^{2}\big{\\}}_{n=0}^{\infty}$ after the latter sequence is
arranged in increasing order. Similarly, the sequence
$\\{H_{m}(-\nu-1,0)\\}_{m=0}^{\infty}$ agrees with
$\big{\\{}(2n-\nu)^{2}\big{\\}}_{n=0}^{\infty}$ arranged in increasing order.
Analysis of this order at $k=0$ implies (a).

(b) If $p=0$ then $\nu=-\frac{1}{2}$ and (b) is trivially true because
$\nu=-\nu-1$. For $p\geq 1$ statement (b) follows from continuity of the
functions $\nu\mapsto H(\nu,k)$ and taking one-sided limits as
$\nu\to-p-\frac{1}{2}$ in (a). ∎

We now compare the eigenvalues $H^{(1)}_{m}(\nu)$ with $H^{(2)}_{m}(\nu)$.

######  Theorem 5.4.

Let $\nu\in\mathbb{R}$, $0<k<1$ and $H_{m}^{(j)}:=H_{m}^{(j)}(\nu,k)$.

  1. $(a)$

If $\nu>-\frac{3}{2}$ then

$\displaystyle
H^{(1)}_{0}<H^{(2)}_{0}<H^{(1)}_{1}<H^{(2)}_{1}<H^{(1)}_{2}<H^{(2)}_{2}<\cdots.$

  2. $(b)$

If $-p-\frac{3}{2}<\nu<-p-\frac{1}{2}$ with $p\in\mathbb{N}$ then

$\displaystyle\left\\{\begin{matrix}H^{(1)}_{0}\\\
H^{(2)}_{0}\end{matrix}\right\\}<\dots<\left\\{\begin{matrix}H^{(1)}_{p-1}\\\
H^{(2)}_{p-1}\end{matrix}\right\\}<H^{(1)}_{p}<H^{(2)}_{p}<H^{(1)}_{p+1}<H^{(2)}_{p+1}<\cdots,$

where, for $m=0,1,\dots,p-1$,

$\displaystyle H^{(1)}_{m}<H^{(2)}_{m}\quad\text{if $m+p$ is even},\qquad
H^{(1)}_{m}>H^{(2)}_{m}\quad\text{if $m+p$ is odd}.$

  3. $(c)$

If $\nu=-p-\tfrac{1}{2}$ with $p\in\mathbb{N}$ then

$\displaystyle
H^{(1)}_{0}=H^{(2)}_{0}<H^{(1)}_{1}=H^{(2)}_{2}<\dots<H^{(1)}_{p-1}=H^{(2)}_{p-1}<H^{(1)}_{p}<H^{(2)}_{p}<\cdots.$

###### Proof.

(a) Let $\nu>-\frac{3}{2}$. Then the eigenfunctions of the two Lamé–Wangerin
problems are constant multiples of solution (3.4) with $q_{0}\neq 0$.
Therefore, the eigenvalues of the two problems are mutually distinct. By
continuity of the functions $k\mapsto H_{m}^{(j)}(\nu,k)$ (Theorem 3.3) the
order of the eigenvalues $H_{m}^{(j)}$ must be the same for all $k\in[0,1)$.
Now

$\displaystyle H^{(1)}_{m}(\nu,0)=(2m+\nu+1)^{2},\qquad
H^{(2)}_{m}(\nu,0)=(2m+\nu+2)^{2},$

which implies (a).

(b) Let $-p-\frac{3}{2}<\nu<-p-\frac{1}{2}$ with $p\in\mathbb{N}$. Again, the
eigenvalues of the two Lamé–Wangerin problems are mutually distinct, and the
order of these eigenvalues must be the same for all $k\in[0,1)$. The sequence
$\big{\\{}H_{m}^{(j)}(\nu,0)\big{\\}}_{m=0}^{\infty}$ is the same as
$\\{(2n+\nu+j)^{2}\\}_{n=0}^{\infty}$ but the latter one has to be ordered
increasingly. An analysis of the order leads to the arrangement stated in (b).

(c) Let $\nu=-p-\tfrac{1}{2}$ with $p\in\mathbb{N}$. Continuity of the
functions $\nu\mapsto H_{m}^{(j)}(\nu,k)$ and part (b) show that
$H^{(1)}_{m}=H^{(2)}_{m}$ for $m=0,1,\dots,p-1$. We know from Theorem 5.3(b)
that

$\displaystyle H^{(j)}_{m+p}(\nu)=H_{m}^{(j)}(-\nu-1),\qquad m\geq 0.$

Since $-\nu-1>-\frac{3}{2}$ the rest of statement (c) follows from part (a). ∎

##  6 Algebraic Lamé functions

If $\nu+\frac{1}{2}$ is a nonzero integer then Lamé’s differential equation
(1.1) has solutions in finite terms which are usually called algebraic Lamé
functions. These solutions were investigated in [4, 7, 8, 13]. We obtain these
functions as follows.

Let $\nu=-p-\tfrac{1}{2}$ with $p\in\mathbb{N}$. For $j=1,2$ we introduce the
symmetric tridiagonal $p$ by $p$ matrices

$\displaystyle S_{p}^{(j)}=\begin{pmatrix}\epsilon_{0}^{(j)}&\delta_{1}&0&&\\\
\delta_{1}&\epsilon^{(j)}_{1}&\ddots&\ddots&\\\ 0&\ddots&\ddots&\ddots&0\\\
&\ddots&\ddots&\epsilon^{(j)}_{p-2}&\delta_{p-1}\\\
&&0&\delta_{p-1}&\epsilon^{(j)}_{p-1}\end{pmatrix},$

where

$\displaystyle\epsilon_{n}^{(j)}=\tfrac{1}{2}k^{2}\big{(}p^{2}-\tfrac{1}{4}\big{)}+(-1)^{j}k^{\prime}(2n+1-p)+\big{(}1-\tfrac{1}{2}k^{2}\big{)}\big{(}\tfrac{1}{4}+(2n+1-p)^{2}\big{)},$
$\displaystyle\delta_{n}=k^{2}n(p-n).$

The coefficient $\delta_{p}$ vanishes in (3.12), (3.19). Therefore, if
$(a_{0},a_{1},\dots,a_{p-1})^{t}$ is an eigenvector of $S^{(j)}_{p}$ and
$a_{n}:=0$ for $n\geq p$ then (3.10), (3.18) are Lamé–Wangerin functions of
the first and second kind, respectively.

We note that $S_{p}^{(1)}$ is the mirror image of $S_{p}^{(2)}$ with respect
to the anti-diagonal, that is, we have

$\displaystyle\epsilon^{(1)}_{n}=\epsilon^{(2)}_{p-1-n},\qquad\delta_{n}=\delta_{p-n}.$

It follows that $S_{p}^{(1)}$ and $S_{p}^{(2)}$ have the same eigenvalues and
the corresponding eigenvectors are inverse to each other, that is, if
$(a_{0},a_{1},{\dots},a_{p-1})^{t}$ is an eigenvector for $S_{p}^{(1)}$ then
$(a_{p-1},a_{p-2},{\dots},a_{0})^{t}$ is an eigenvector for $S_{p}^{(2)}$
belonging to the same eigenvalue. According to Theorem 5.4(c), the common
eigenvalues of $S_{p}^{(j)}$ are

$\displaystyle
H_{m}^{(1)}(-p-\tfrac{1}{2})=H_{m}^{(2)}\big{(}{-}p-\tfrac{1}{2}\big{)},\qquad
m=0,1,\dots,p-1.$

If $(a_{0},a_{1},\dots,a_{p-1})^{t}$ is a (real) eigenvector of $S_{p}^{(1)}$
then

$\displaystyle
w^{(1)}=\eta^{-\frac{1}{2}p+\frac{1}{4}}(\eta_{2}-\eta)^{1/2}\sum_{n=0}^{p-1}a_{n}\eta^{n},$
(6.1) $\displaystyle
w^{(2)}=\eta^{-\frac{1}{2}p+\frac{1}{4}}(\eta_{1}-\eta)^{1/2}\sum_{n=0}^{p-1}a_{p-n-1}\eta^{n}$
(6.2)

are solutions of (3.11). These are algebraic Lamé functions expressed in the
variable $\eta$. We note that the functions $w^{(1)}$ and $w^{(2)}$ are
essentially Heun polynomials. For if we set
$w=\eta^{-\frac{1}{2}p+\frac{1}{4}}(\eta_{j}-\eta)^{1/2}v(s)$ and
$\eta=\eta_{1}s$, then we obtain the Heun equation for $v(s)$ and
$\sum\limits_{n=0}^{p-1}a_{n}(\eta_{1}s)^{n}$ is a Heun polynomial.

If we substitute (3.7) in (6.1), (6.2) and use the functions $J_{1}(z)$,
$J_{2}(z)$ defined in (4.7), (4.8) we obtain

$\displaystyle
w^{(1)}(z)=J_{1}(z)\sum_{n=0}^{p-1}a_{n}(\operatorname{sn}z-{\rm
i}\operatorname{cn}z)^{2n-p+1},$ $\displaystyle
w^{(2)}(z)=J_{2}(z)\sum_{n=0}^{p-1}a_{n}(\operatorname{sn}z+{\rm
i}\operatorname{cn}z)^{2n-p+1}.$

We know from Lemma 5.1 that

$\displaystyle w^{(1)}(z+2K)={\rm i}(-1)^{p}w^{(1)}(z),\qquad
w^{(2)}(z+2K)={\rm i}(-1)^{p+1}w^{(2)}(z).$

Moreover, we have

$\displaystyle(1+k^{\prime})^{1/2}\overline{w^{(1)}(\bar{z})}=-{\rm
i}(1-k^{\prime})^{1/2}w^{(2)}(z),$

and, for $x\in\mathbb{R}$,

$\displaystyle\overline{w^{(1)}(x)}=w^{(1)}(2K-x),\qquad\overline{w^{(2)}(x)}=-w^{(2)}(2K-x),$

which shows that the real part of $w^{(1)}(x)$ is a function even with respect
to $x=K$ while the imaginary part of $w^{(1)}(x)$ is odd with respect to
$x=K$.

One should notice that if $\nu=-p-\frac{1}{2}$ and $h$ is an eigenvalue of the
matrix $S_{p}^{(j)}$ then all (nontrivial) solutions of Lamé’s equation
qualify as “algebraic Lamé functions”. We picked the basis of two solutions,
one even and one odd with respect to $z=K+{\rm i}K^{\prime}$. Ince [8]
considered the basis of even and odd solutions (with respect to $z=0$) while
Erdélyi [4] has the basis of even or odd solutions with respect to $z=K$.

In the simplest case $\nu=-\frac{3}{2}$ we have

$\displaystyle
H_{0}^{(1)}\big{(}{-}\tfrac{3}{2}\big{)}=H_{0}^{(2)}\big{(}{-}\tfrac{3}{2}\big{)}=\tfrac{1}{4}\big{(}1+k^{2}\big{)}$

and

$\displaystyle w_{0}^{(1)}(z)=J_{1}(z),\qquad w_{0}^{(2)}(z)=J_{2}(z).$

If $\nu=-\frac{5}{2}$ then

$\displaystyle
H_{0}^{(1)}\big{(}{-}\tfrac{5}{2}\big{)}=H_{0}^{(2)}\big{(}{-}\tfrac{5}{2}\big{)}=\tfrac{5}{4}\big{(}1+k^{2}\big{)}-\big{(}1-k^{2}+k^{4}\big{)}^{1/2},$
$\displaystyle
H_{1}^{(1)}\big{(}{-}\tfrac{5}{2}\big{)}=H_{1}^{(2)}\big{(}{-}\tfrac{5}{2}\big{)}=\tfrac{5}{4}\big{(}1+k^{2}\big{)}+\big{(}1-k^{2}+k^{4}\big{)}^{1/2}.$

If we choose $a_{0}=-k^{2}$,
$a_{1}=\frac{3}{4}k^{2}+\frac{9}{4}-\frac{1}{2}k^{2}\eta_{1}-H_{m}^{(1)}\big{(}{-}\frac{5}{2}\big{)}$,
$m=0,1$, then

$\displaystyle w_{m}^{(1)}(z)=J_{1}(z)(a_{0}(\operatorname{sn}z+{\rm
i}\operatorname{cn}z)+a_{1}(\operatorname{sn}z-{\rm i}\operatorname{cn}z)),$
$\displaystyle w_{m}^{(2)}(z)=J_{2}(z)(a_{0}(\operatorname{sn}z-{\rm
i}\operatorname{cn}z)+a_{1}(\operatorname{sn}z+{\rm i}\operatorname{cn}z)).$

##  7 Lamé polynomials

Let $\nu=-p-1$ with $p\in\mathbb{N}_{0}$. It is well-known [1, Chapter IX]
that there are $2p+1$ distinct values of $h$ for which (1.1) admits nontrivial
solutions which are polynomials in $\operatorname{cn}z$, $\operatorname{sn}z$,
$\operatorname{dn}z$. In our notation these values of $h$ are

$\displaystyle H_{m}^{(1)}(-p-1),\qquad m=0,1,\dots,p$ (7.1)

and

$\displaystyle H_{m}^{(2)}(-p-1),\qquad m=0,1,\dots,p-1.$ (7.2)

Since $\alpha_{p+1}=0$ in (3.9), the numbers (7.1) are the eigenvalues of the
$p+1$ by $p+1$ tridiagonal matrix

$\displaystyle T_{p+1}^{(1)}=\begin{pmatrix}\beta_{0}&\gamma_{1}&0&&\\\
\alpha_{1}&\beta_{1}&\ddots&\ddots&\\\ 0&\ddots&\ddots&\ddots&0\\\
&\ddots&\ddots&\beta_{p-1}&\gamma_{p}\\\
&&0&\alpha_{p}&\beta_{p}\end{pmatrix},$

where

$\displaystyle\alpha_{n}=\tfrac{1}{2}k^{2}(p+1-n)(2n-1),$
$\displaystyle\beta_{n}=\tfrac{1}{2}k^{2}p(p+1)+\big{(}1-\tfrac{1}{2}k^{2}\big{)}(2n-p)^{2},$
$\displaystyle\gamma_{n}=\tfrac{1}{2}k^{2}(2p+1-2n)n.$

If $(c_{0},c_{1},\dots,c_{p})^{t}$ is an eigenvector of $T_{p+1}^{(1)}$ then

$\displaystyle w=\eta^{-p/2}\sum_{n=0}^{p}c_{n}\eta^{n}$

is a solution of (3.6). After substituting (3.7) we obtain

$\displaystyle w=\sum_{n=0}^{p}c_{n}(\operatorname{sn}z-{\rm
i}\operatorname{cn}z)^{2n-p}.$

Indeed, since $(\operatorname{sn}z-{\rm
i}\operatorname{cn}z)^{-1}=\operatorname{sn}z+{\rm i}\operatorname{cn}z$,
these solutions are polynomials in $\operatorname{cn}z$, $\operatorname{sn}z$.
The matrix $T_{p+1}^{(1)}$ has the symmetries

$\displaystyle\alpha_{n}=\gamma_{p-n+1},\qquad\beta_{n}=\beta_{p-n}.$

Therefore, the space of symmetric vectors $\\{c_{n}\\}_{n=0}^{p}$
($c_{n}=c_{p-n}$, $n=0,1,\dots,p$), as well as the space of antisymmetric
vectors is invariant under $T_{p+1}^{(1)}$. Thus eigenvectors of
$T_{p+1}^{(1)}$ will lie in one of these invariant subspaces.

If $p$ is even we find $\tfrac{1}{2}p+1$ Lamé polynomials of the form
$P\big{(}\operatorname{sn}^{2}z\big{)}$ where $P$ is a polynomial of degree
$\tfrac{1}{2}p$ if we use symmetric eigenvectors, and $\tfrac{1}{2}p$ Lamé
polynomials of the form
$\operatorname{cn}z\operatorname{sn}zP\big{(}\operatorname{sn}^{2}z\big{)}$
where $P$ is a polynomial of degree $\tfrac{1}{2}p-1$ if we use antisymmetric
eigenvectors. If $p$ is odd we find $\tfrac{1}{2}(p+1)$ Lamé polynomials of
the form $\operatorname{sn}zP\big{(}\operatorname{sn}^{2}z\big{)}$ where $P$
is a polynomial of degree $\tfrac{1}{2}(p-1)$ if we use symmetric
eigenvectors, and $\tfrac{1}{2}(p+1)$ Lamé polynomials of the form
$\operatorname{cn}zP\big{(}\operatorname{sn}^{2}z\big{)}$ where $P$ is a
polynomial of degree $\tfrac{1}{2}(p-1)$ if we use antisymmetric eigenvectors.

Similarly, Lamé–Wangerin functions of the second kind belonging to the
eigenvalues (7.2) are Lamé polynomials that have the factor
$\operatorname{dn}z$.

##  8 Zeros of Lamé–Wangerin functions

We first determine the number of zeros of Lamé–Wangerin functions
$w_{m}^{(j)}(z)$ in the open interval $({\rm i}K^{\prime},K+{\rm
i}K^{\prime})$.

######  Theorem 8.1.

Let $j=1,2$, $m\in\mathbb{N}_{0}$, $\nu\in\mathbb{R}$, $k\in(0,1)$.

  1. $(a)$

If $\nu>-\frac{3}{2}$ then $w_{m}^{(j)}$ has exactly $m$ zeros in $({\rm
i}K^{\prime},K+{\rm i}K^{\prime})$.

  2. $(b)$

If $-p-\tfrac{3}{2}<\nu\leq-p-\tfrac{1}{2}$, $p\in\mathbb{N}$, then
$w_{m}^{(j)}$ has exactly $\max\\{0,m-p\\}$ zeros in $({\rm
i}K^{\prime},K+{\rm i}K^{\prime})$.

###### Proof.

Consider $j=1$. The proof for $j=2$ similar. Let $P$ be the set of all real
numbers different from $-p-\frac{1}{2}$ for all $p\in\mathbb{N}$. For
$h\in\mathbb{R}$, $\nu\in P$, let $w(z,h,\nu)$ be the solution of (1.1) given
locally at $z={\rm i}K^{\prime}$ by (3.4) with $q_{0}=1$. Then one can show
that $(z-{\rm i}K^{\prime})^{-\nu-1}w(z,h,\nu)$ is continuous for $z$ in
$[{\rm i}K^{\prime},K+{\rm i}K^{\prime}]$, $h\in\mathbb{R}$, $\nu\in P$ ($k$
fixed). If we set $w_{m}(z,\nu)=w\big{(}z,H^{(1)}_{m}(\nu,k),\nu\big{)}$ then
$(z-{\rm i}K^{\prime})^{-\nu-1}w_{m}(z,\nu)$ is continuous for $z\in[{\rm
i}K^{\prime},K+{\rm i}K^{\prime}]$ and $\nu\in P$. This implies that the
number of zeros of $w_{m}(\cdot,\nu)$ in $({\rm i}K^{\prime},K+{\rm
i}K^{\prime})$ is finite and it is locally constant as a function of $\nu$.

(a) follows by considering $\nu=0$:

$\displaystyle
w_{m}^{(1)}(z,0,k)=(-1)^{m}\sin\left((2m+1)\frac{\pi}{2K}(z-{\rm
i}K^{\prime})\right).$ (8.1)

(b) Suppose $-p-\tfrac{3}{2}<\nu\leq-p-\tfrac{1}{2}$ with $p\in\mathbb{N}$.
Let $m=0,1,\dots,p$. By Theorem 5.3(a), we have $H_{m}(\nu)\leq
H_{0}(-\nu-1)$. By (a), $w_{0}(\cdot,-\nu-1)$ has no zeros in $({\rm
i}K^{\prime},K+{\rm i}K^{\prime})$. Therefore, by Sturm comparison,
$w_{m}(\cdot,\nu)$ also has no zeros in this interval.

Now consider $m>p$. If $\nu=-p-\frac{1}{2}$ then, by Theorem 5.3(b),
$H_{m}(\nu)=H_{m-p}(-\nu-1)$. Therefore, by (a), $w_{m}(\cdot,\nu)$ has $m-p$
zeros in $({\rm i}K^{\prime},K+{\rm i}K^{\prime})$. If
$-p-\frac{3}{2}<\nu<p-\frac{1}{2}$ then, by Theorem 5.3(a),
$H_{m}(\nu)<H_{m-p}(-\nu-1)$. Therefore, $w_{m}(\cdot,\nu)$ can have at most
$m-p$ zeros in $({\rm i}K^{\prime},K+{\rm i}K^{\prime})$. If
$\nu=-p-\frac{1}{2}$ we just showed that there are $m-p$ zeros. By continuity,
there are exactly $m-p$ zeros. For the latter step Lamé–Wangerin functions
should be normalized by the initial conditions $w(K+{\rm i}K^{\prime})=1$,
$w^{\prime}(K+{\rm i}K^{\prime})=0$. ∎

Now we look for zeros of Lamé–Wangerin functions in the strip $0\leq\Im
z<K^{\prime}$.

######  Lemma 8.2.

A Lamé–Wangerin function which is not a Lamé polynomial has no zeros on the
real axis.

###### Proof.

If $\mu$ is not an integer then a nontrivial Floquet solution $w(z)$,
$z\in\mathbb{R}$, of (1.1) with $w(z+2K)={\rm e}^{{\rm i}\mu\pi}w(z)$ does not
have zeros on the real axis. This is because the conjugate of $w(z)$ is a
Floquet solution with conjugate multiplier ${\rm e}^{-{\rm i}\mu\pi}$, and
${\rm e}^{{\rm i}\mu\pi}$, ${\rm e}^{-{\rm i}\mu\pi}$ are distinct. So $w(z)$
and its conjugate function are linearly independent. It follows from Lemma 5.1
that Lamé–Wangerin functions have no zeros on the real axis if $\nu$ is not an
integer.

Suppose that $\nu$ is an integer, and $w(z)$ is a Lamé–Wangerin function
belonging to the eigenvalue $H^{(1)}_{m}(\nu)$. Suppose that $w(z_{0})=0$.
with $z_{0}\in\mathbb{R}$. Using (3.8) and the substitution (2.4) we have

$\displaystyle w(t)=\sum_{n=0}^{\infty}c_{n}{\rm e}^{-{\rm i}t(2n+\nu+1)}$

and this function has a zero at $t_{0}\in\mathbb{R}$. The coefficients $c_{n}$
are real so the functions

$\displaystyle\Re w(t)=\sum_{n=0}^{\infty}c_{n}\cos(2n+\nu+1)t,\qquad
t\in\mathbb{R}$ (8.2) $\displaystyle\Im
w(t)=-\sum_{n=0}^{\infty}c_{n}\sin(2n+\nu+1)t,\qquad t\in\mathbb{R}$ (8.3)

both vanish at $t=t_{0}$. The functions (8.2), (8.3) are both solutions of the
differential equation (2.5) with the same values for $h$ and $\nu$. Since they
have a common zero these solutions must be linearly dependent. Now $\Re w(t)$
is an even function and $\Im w(t)$ is odd. So one of the functions $\Re w(t)$,
$\Im w(t)$ must vanish identically. This implies that $c_{n}=0$ for large
enough $n$ and so $w(z)$ is a Lamé polynomial. The proof is similar for
Lamé–Wangerin function of the second kind. ∎

According to (3.10) we write a Lamé–Wangerin function of the first kind as

$\displaystyle
w_{m}^{(1)}=\eta^{(\nu+1)/2}(\eta_{2}-\eta)^{1/2}v^{(1)}_{m}(\eta,\nu,k),$

where

$\displaystyle v_{m}^{(1)}(\eta,\nu,k)=\sum_{n=0}^{\infty}a_{n}\eta^{n}$

is given by a power series with radius $\geq\eta_{2}>1$. Similarly, we write a
Lamé–Wangerin function of the second kind as

$\displaystyle
w_{m}^{(2)}=\eta^{(\nu+1)/2}(\eta_{1}-\eta)^{1/2}v^{(2)}_{m}(\eta,\nu,k).$

######  Theorem 8.3.

Let $m\in\mathbb{N}_{0}$, $\nu\in\mathbb{R}$, $k\in(0,1)$.

  1. $(a)$

Suppose that $-m-\nu\not\in\mathbb{N}$, and choose $\ell\in\mathbb{N}_{0}$
such that $H^{(1)}_{m}(\nu,0)=(2\ell+\nu+1)^{2}$; see Lemma 3.2. Then
$v_{m}^{(1)}(\cdot,\nu,k)$ has exactly $\ell$ zeros in the unit disk
$|\eta|<1$ counted by multiplicity.

  2. $(b)$

Suppose that $-m-\nu-1\not\in\mathbb{N}$, and choose $\ell\in\mathbb{N}_{0}$
such that $H^{(2)}(\nu,0)=(2\ell+\nu+2)^{2}$. Then $v_{m}^{(2)}(\cdot,\nu,k)$
has exactly $\ell$ zeros in the unit disk $|\eta|<1$ counted by multiplicity.

###### Proof.

We prove only (a). The proof of (b) is similar. We normalize the Lamé–Wangerin
functions $w_{m}(z)$ of the first kind by setting $w_{m}(K+{\rm
i}K^{\prime})=1$. Then $w_{m}(z,\nu,k)$ is the solution of (1.1) with
$h=H^{(1)}_{m}(\nu,k)$ determined by the initial conditions $w(K+{\rm
i}K^{\prime})=1$, $w^{\prime}(K+{\rm i}K^{\prime})=0$. By continuous parameter
dependence of solutions of initial value problems of linear differential
equations, and using Theorem 3.3, we obtain that $w_{m}(z,\nu,k)$ is a
continuous function of $(z,\nu,k)$ for $z\in\mathbb{R}$, $\nu\in\mathbb{R}$,
$k\in(0,1)$. Since $|\eta|=1$ is in correspondence with $z\in[0,2K)$, we see
that $v_{m}(\eta,\nu,k)$ is a continuous function of $|\eta|=1$,
$\nu\in\mathbb{R}$, $k\in(0,1)$. We want to apply Rouché’s theorem to the
homotopy $s\mapsto v_{m}(\eta,s\nu,k)$ for $s\in[0,1]$. If
$v_{m}(\eta,s\nu,k)\neq 0$ on the unit circle $|\eta|=1$ for all $s\in[0,1]$,
then $v_{m}(\cdot,s\nu,k)$ has the same number of zeros in $|\eta|<1$ for
$s\in[0,1]$.

Suppose that $\nu>-m-1$. By Lemma 8.2, the function $v_{m}(\eta,s\nu,k)$ has
no zeros on the unit circle $|\eta|=1$ for $0\leq s\leq 1$ and so the number
of zeros of $v_{m}(\cdot,\nu,k)$ in the open unit disk agrees with that of
$v_{m}(\cdot,0,k)$. It follows from (8.1) that the number of zeros of
$v_{m}(\cdot,\nu,k)$ in the open unit disk is equal to $m$. Under our
assumption on $(\nu,m)$ we have $\ell=m$, so we obtain statement (a) for
$\nu>-m-1$.

Now we assume that $-p-1<\nu<-p$ with $p\in\mathbb{N}$ and $m<p$. We use
similar homotopies to show that the number of zeros of $v_{m}(\cdot,\nu,k)$
may depend on $p$ and $m$ but not on $\nu,k\in(0,1)$. So we consider
$\nu=-p-\frac{1}{2}$. Then $w$ is an algebraic Lamé function and
$v_{m}(\eta)=\sum\limits_{n=0}^{p-1}a_{n}\eta^{n}$ is a polynomial. Let
$k_{n}\in(0,1)$ be a sequence converging to $0$. Since the vector
$(a_{0},a_{1},\dots,a_{p-1})^{t}$ is an eigenvector of the matrix
$S_{p}^{(1)}$ from Section 6 it is easy to see that when properly normalized
the eigenvectors belonging to $k_{n}$ converge to the vector
$(a_{0},\dots,a_{p-1})^{t}$ with all components equal to $0$ except
$a_{\ell}=1$. Therefore, under the new normalization
$v_{m}\big{(}\eta,-p-\frac{1}{2},k_{n}\big{)}$ converges uniformly to
$\eta^{\ell}$ as $n\to 0^{+}$. By Rouché’s theorem, we obtain the desired
statement.

This completes the proof. ∎

Using the map (3.7) the unit disk $|\eta|<1$ can be related to a domain in the
$z$-plane. Consider the rectangle

$\displaystyle Q=\\{z\in\mathbb{C}\colon 0<\Re z<2K,\,0<\Im z<K^{\prime}\\}.$

The function $z\mapsto\eta$ is a conformal map from $Q$ onto the unit disk
$|\eta|<1$ with a branch cut along the interval $(-1,\eta_{1})$. If $z$ starts
at $z=0$ and moves clockwise around the boundary of $Q$, then $\eta$ starts at
$\eta=-1$ and moves in the mathematically positive direction along the unit
circle returning to $\eta=-1$ when $z=2K$. Then $\eta$ moves from $\eta=-1$ to
$\eta=0$ when $z$ reaches $z={\rm i}K^{\prime}$. Then $\eta$ moves to
$\eta_{1}$ when $z=K+{\rm i}K^{\prime}$ and returns to $\eta=0$, then to
$\eta=-1$. It follows that the set

$\displaystyle\tilde{Q}=\\{z\colon 0\leq\Re z\leq K,\,0<\Im z\leq
K^{\prime}\\}\cup\\{z\colon K<\Re z<2K,\,0<\Im z<K^{\prime}\\}$

is mapped bijectively onto the open unit disk $|\eta|<1$.

Theorem 8.3(a) can be extended to include Lamé polynomials. If
$\nu=-m-1,-m-2,\dots$ then let $\ell_{1}$ be the smallest nonnegative integer
$n$ satisfying $H^{(1)}_{m}(\nu,0)=(2n+\nu+1)^{2}$ and $\ell_{2}$ the largest
such integer. Then $v_{m}$ has $\ell_{1}$ zeros in the open unit disk
$|\eta|<1$ and $\ell_{2}$ zeros in the closed unit disk $|\eta|\leq 1$. This
follows from the known location of zeros of Lamé polynomials [1, Section 9.4].
Similarly, Theorem 8.3(b) can be extended.

##  9 The limit $\boldsymbol{k\to 0}$ of Lamé–Wangerin functions

Substituting $u=\frac{2K}{\pi}s$ in (3.1), we obtain the differential equation

$\displaystyle\frac{{\rm d}^{2}w}{{\rm
d}s^{2}}+\frac{4K^{2}}{\pi^{2}}\left(h-\nu(\nu+1)\operatorname{sn}^{-2}\left(\frac{2K}{\pi}s\right)\right)w=0,\qquad
0<s<\pi.$ (9.1)

In (9.1) we set $h=H^{(1)}_{m}(\nu,k)$ and take $w=w_{m}^{(1)}(s,\nu,k)$ as
the corresponding Lamé–Wangerin eigenfunction normalized by the initial
condition

$\displaystyle w\left(\frac{\pi}{2}\right)=1,\qquad\frac{{\rm d}w}{{\rm
d}s}\left(\frac{\pi}{2}\right)=0.$

By Theorem 3.3, $H^{(1)}_{m}(\nu,k)\to H^{(1)}_{m}(\nu,0)=(2\ell+\nu+1)^{2}$
as $k\to 0^{+}$, where $\ell\in\mathbb{N}_{0}$ is chosen according to Lemma
3.2. As $k\to 0^{+}$ we see that $w_{m}^{(1)}(s,\nu,k)$ converges to the
solution $W^{(1)}_{m}(s,\nu)$ of the differential equation

$\displaystyle\frac{{\rm d}^{2}W}{{\rm
d}s^{2}}+\left((2\ell+\nu+1)^{2}-\frac{\nu(\nu+1)}{\sin^{2}s}\right)W=0$ (9.2)

satisfying the initial conditions

$\displaystyle W\left(\frac{\pi}{2}\right)=1,\qquad\frac{{\rm d}W}{{\rm
d}s}\left(\frac{\pi}{2}\right)=0.$

The convergence

$\displaystyle w_{m}^{(1)}(s,\nu,k)\to W_{m}^{(1)}(s,\nu)\qquad\text{as $k\to
0^{+}$}$

is uniform on compact subintervals of $(0,\pi)$. Differential equation (9.2)
appears in the theory of Gegenbauer polynomials [20, equation (4.7.11)]. We
find that

$\displaystyle W_{m}^{(1)}(s,\nu)=(\sin
s)^{\nu+1}F\big{(}{-}\ell,\ell+\nu+1;\tfrac{1}{2};\cos^{2}s\big{)},$

where $F$ denotes the hypergeometric function. Equivalently, using Gegenbauer
polynomials $G_{n}^{(\lambda)}(x)$ we have [20, equation (4.7.30)]

$\displaystyle W_{m}^{(1)}(s,\nu)=(\sin
s)^{\nu+1}(-1)^{\ell}\binom{\ell+\nu}{\ell}^{-1}G_{2\ell}^{(\nu+1)}(\cos s).$

The binomial coefficient may vanish but the formula remains valid if we take
limits $\nu\to\nu_{0}$ at exceptional values $\nu=\nu_{0}$.

Similarly, let $w^{(2)}_{m}(s,\nu,k)$ be the solution of (9.1) with
$h=H_{m}^{(2)}(\nu,k)$ satisfying the initial conditions

$\displaystyle w\left(\frac{\pi}{2}\right)=0,\qquad\frac{{\rm d}w}{{\rm
d}s}\left(\frac{\pi}{2}\right)=1.$

We choose $\ell\in\mathbb{N}_{0}$ such that
$H^{(2)}_{m}(\nu,0)=(2\ell+\nu+2)^{2}$. Then we obtain

$\displaystyle w_{m}^{(2)}(s,\nu,k)\to W^{(2)}_{m}(s,\nu)=-(\sin
s)^{\nu+1}\cos sF\big{(}{-}\ell,\ell+\nu+2;\tfrac{3}{2};\cos^{2}s\big{)}$

as $k\to 0^{+}$ uniformly on compact subintervals of $(0,\pi)$. In terms of
Gegenbauer polynomials we have

$\displaystyle W^{(2)}_{m}(s,\nu)=(\sin
s)^{\nu+1}(-1)^{\ell+1}\left(2(\nu+1)\binom{\ell+\nu+1}{\ell}\right)^{-1}G_{2\ell+1}^{(\nu+1)}(\cos
s).$

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