# Fourier and Gegenbauer expansions for a fundamental solution of the
Laplacian in the hyperboloid model of hyperbolic geometry

H S Cohl1,2 and E G Kalnins3 1Information Technology Laboratory, National
Institute of Standards and Technology, Gaithersburg, MD, USA  2Department of
Mathematics, University of Auckland, Auckland, New Zealand  3Department of
Mathematics, University of Waikato, Hamilton, New Zealand<EMAIL_ADDRESS>
###### Abstract

Due to the isotropy $d$-dimensional hyperbolic space, there exist a
spherically symmetric fundamental solution for its corresponding Laplace-
Beltrami operator. On the $R$-radius hyperboloid model of $d$-dimensional
hyperbolic geometry with $R>0$ and $d\geq 2$, we compute azimuthal Fourier
expansions for a fundamental solution of Laplace’s equation. For $d\geq 2$, we
compute a Gegenbauer polynomial expansion in geodesic polar coordinates for a
fundamental solution of Laplace’s equation on this negative-constant sectional
curvature Riemannian manifold. In three-dimensions, an addition theorem for
the azimuthal Fourier coefficients of a fundamental solution for Laplace’s
equation is obtained through comparison with its corresponding Gegenbauer
expansion.

###### pacs:

02.30.Em, 02.30.Gp, 02.30.Jr, 02.30.Nw, 02.40.Ky, 02.40.Vh

###### ams:

31C12, 32Q45, 33C05, 33C45, 35A08, 35J05, 42A16

##  1 Introduction

In this paper we discuss eigenfunction expansions for a fundamental solution
of Laplace’s equation in the hyperboloid model of $d$-dimensional hyperbolic
geometry. In particular, for a fixed $R\in(0,\infty)$ and $d\geq 2$, we derive
and discuss Fourier cosine and Gegenbauer polynomial expansions in
rotationally invariant coordinate systems, for a previously derived (see Cohl
& Kalnins (2011) [8]) spherically symmetric fundamental solution of the
Laplace-Beltrami operator in the hyperboloid model of hyperbolic geometry.
Useful background material relevant for this paper can be found in Vilenkin
(1968) [29], Thurston (1997) [27], Lee (1997) [20] and Pogosyan & Winternitz
(2002) [26].

This paper is organized as follows. In section 2, for the hyperboloid model of
$d$-dimensional hyperbolic geometry, we describe some of its global
properties, such as its geodesic distance function, geodesic polar
coordinates, Laplace-Beltrami operator (Laplacian), radial harmonics, and
several previously derived equivalent expressions for a radial fundamental
solution of Laplace’s equation. In section 3, for $d\geq 2$, we derive and
discuss Fourier cosine series for a fundamental solution of Laplace’s equation
about an appropriate azimuthal angle in rotationally invariant coordinate
systems, and show how the resulting Fourier coefficients compare to the those
in Euclidean space. In section 4, for $d\geq 2$, we compute Gegenbauer
polynomial expansions in geodesic polar coordinates, for a fundamental
solution of Laplace’s equation in the hyperboloid model of hyperbolic
geometry. In section 5 we discuss possible directions of research in this
area.

Throughout this paper we rely on the following definitions. For
$a_{1},a_{2},\ldots\in{\mathbf{C}}$, if $i,j\in{\mathbf{Z}}$ and $j<i$ then
$\sum_{n=i}^{j}a_{n}=0$ and $\prod_{n=i}^{j}a_{n}=1$. The set of natural
numbers is given by ${\mathbf{N}}:=\\{1,2,\ldots\\}$, the set
${\mathbf{N}}_{0}:=\\{0,1,2,\ldots\\}={\mathbf{N}}\cup\\{0\\}$, and the set
${\mathbf{Z}}:=\\{0,\pm 1,\pm 2,\ldots\\}.$ The set ${\mathbf{R}}$ represents
the real numbers.

##  2 Global analysis on the hyperboloid

###  2.1 The hyperboloid model of hyperbolic geometry

Hyperbolic space, developed independently by Lobachevsky and Bolyai around
1830 (see Trudeau (1987) [28]), is a fundamental example of a space exhibiting
hyperbolic geometry. Hyperbolic geometry is analogous to Euclidean geometry,
but such that Euclid’s parallel postulate is no longer assumed to hold. There
are several models of $d$-dimensional hyperbolic space ${\mathbf{H}}_{R}^{d}$,
including the Klein, Poincaré, hyperboloid, upper-half space and hemisphere
models (see Thurston (1997) [27]). The hyperboloid model for $d$-dimensional
hyperbolic geometry (hereafter referred to as the hyperboloid model, or more
simply, the hyperboloid), is closely related to the Klein and Poincaré models:
each can be obtained projectively from the others. The upper-half space and
hemisphere models can be obtained from one another by inversions with the
Poincaré model (see section 2.2 in Thurston (1997) [27]). The model of
hyperbolic geometry which we will be focusing on in this paper, is the
hyperboloid model.

Minkowski space ${\mathbf{R}}^{d,1}$ is a $(d+1)$-dimensional pseudo-
Riemannian manifold which is a real finite-dimensional vector space, with
Cartesian coordinates given by ${\bf x}=(x_{0},x_{1},\ldots,x_{d})$. The
hyperboloid model, also known as the Minkowski or Lorentz models, represents
points in this space by the upper sheet $(x_{0}>0)$ of a two-sheeted
hyperboloid embedded in the Minkowski space ${\mathbf{R}}^{d,1}$. It is
equipped with a nondegenerate, symmetric bilinear form, the Minkowski bilinear
form

$[{\bf x},{\mathbf{y}}]=x_{0}y_{0}-x_{1}y_{1}-\ldots-x_{d}y_{d}.$

The above bilinear form is symmetric, but not positive-definite, so it is not
an inner product. It is defined analogously with the Euclidean inner product
for ${\mathbf{R}}^{d+1}$

$({\bf x},{\mathbf{y}})=x_{0}y_{0}+x_{1}y_{1}+\ldots+x_{d}y_{d}.$

The variety $[{\bf x},{\bf x}]=x_{0}^{2}-x_{1}^{2}-\ldots-x_{d}^{2}=R^{2}$,
for ${\bf x}\in{\mathbf{R}}^{d,1}$, using the language of Beltrami (1869) [3]
(see also p. 504 in Vilenkin (1968) [29]), defines a pseudo-sphere of radius
$R$. Points on the pseudo-sphere with zero radius coincide with a cone. Points
on the pseudo-sphere with radius greater than zero lie within this cone, and
points on the pseudo-sphere with purely imaginary radius lie outside the cone.

For a fixed $R\in(0,\infty),$ the $R$-radius hyperboloid model is a maximally
symmetric, simply connected, $d$-dimensional Riemannian manifold with
negative-constant sectional curvature (given by $-1/R^{2}$, see for instance
p. 148 in Lee (1997) [20]), whereas Euclidean space ${\mathbf{R}}^{d}$
equipped with the Pythagorean norm, is a Riemannian manifold with zero
sectional curvature. The hypersphere ${\mathbf{S}}^{d}$, is an example of a
space (submanifold) with positive-constant sectional curvature (given by
$1/R^{2}$).

In our discussion of a fundamental solution for Laplace’s equation in the
hyperboloid model ${\mathbf{H}}_{R}^{d}$, we focus on the positive radius
pseudo-sphere which can be parametrized through subgroup-type coordinates,
i.e. those which correspond to a maximal subgroup chain $O(d,1)\supset\ldots$
(see for instance Pogosyan & Winternitz (2002) [26]). There exist separable
coordinate systems which parametrize points on the positive radius pseudo-
sphere (i.e. such as those which are analogous to parabolic coordinates, etc.)
which can not be constructed using maximal subgroup chains (we will no longer
discuss these).

Geodesic polar coordinates are coordinates which correspond to the maximal
subgroup chain given by $O(d,1)\supset O(d)\supset\ldots$. What we will refer
to as standard geodesic polar coordinates correspond to the subgroup chain
given by $O(d,1)\supset O(d)\supset O(d-1)\supset\cdots\supset O(2).$ Standard
geodesic polar coordinates (see Olevskiĭ (1950) [23]; Grosche, Pogosyan &
Sissakian (1997) [16]), similar to standard hyperspherical coordinates in
Euclidean space, can be given by

$\left.\begin{array}[]{rcl}x_{0}&=&R\cosh r\\\\[0.56917pt] x_{1}&=&R\sinh
r\cos\theta_{1}\\\\[2.84544pt] x_{2}&=&R\sinh
r\sin\theta_{1}\cos\theta_{2}\\\\[2.84544pt] &\vdots&\\\\[0.56917pt]
x_{d-2}&=&R\sinh r\sin\theta_{1}\cdots\cos\theta_{d-2}\\\\[2.84544pt]
x_{d-1}&=&R\sinh r\sin\theta_{1}\cdots\sin\theta_{d-2}\cos\phi\\\\[2.84544pt]
x_{d}&=&R\sinh
r\sin\theta_{1}\cdots\sin\theta_{d-2}\sin\phi,\end{array}\quad\right\\}$ (1)

where $r\in[0,\infty)$, $\phi\in[0,2\pi)$, and $\theta_{i}\in[0,\pi]$ for
$i\in\\{1,\ldots,d-2\\}$.

In order to study fundamental solutions on the hyperboloid, we need to
describe how one computes distances in this space. One may naturally compare
distances on the positive radius pseudo-sphere through analogy with the
$R$-radius hypersphere. Distances on the hypersphere are simply given by arc
lengths, angles between two arbitrary vectors, from the origin, in the ambient
Euclidean space. We consider the $d$-dimensional hypersphere embedded in
${\mathbf{R}}^{d+1}$. Points on the hypersphere can be parametrized using
hyperspherical coordinate systems. Any parametrization of the hypersphere
${\mathbf{S}}^{d}$, must have $({\bf x},{\bf
x})=x_{0}^{2}+\ldots+x_{d}^{2}=R^{2}$, with $R>0$. The distance between two
points on the hypersphere is given by

$d({\bf x},{{\bf x}^{\prime}})=R\gamma=R\cos^{-1}\left(\frac{({\bf x},{{\bf
x}^{\prime}})}{({\bf x},{\bf x})({{\bf x}^{\prime}},{{\bf
x}^{\prime}})}\right)=R\cos^{-1}\left(\frac{1}{R^{2}}({\bf x},{{\bf
x}^{\prime}})\right).$ (2)

This is evident from the fact that the geodesics on ${\mathbf{S}}^{d}$ are
great circles (i.e. intersections of ${\mathbf{S}}^{d}$ with planes through
the origin) with constant speed parametrizations (see p. 82 in Lee (1997)
[20]).

Accordingly, we now look at the geodesic distance function on the
$d$-dimensional positive radius pseudo-sphere ${\mathbf{H}}_{R}^{d}.$
Distances between two points on the positive radius pseudo-sphere are given by
the hyperangle between two arbitrary vectors, from the origin, in the ambient
Minkowski space. Any parametrization of the hyperboloid
${\mathbf{H}}_{R}^{d}$, must have $[{\bf x},{\bf x}]=R^{2}$. The geodesic
distance $\rho\in[0,\infty)$ between two points ${\bf x},{{\bf
x}^{\prime}}\in{\mathbf{H}}_{R}^{d}$ is given by

$d({\bf x},{{\bf x}^{\prime}})=R\cosh^{-1}\left(\frac{[{\bf x},{{\bf
x}^{\prime}}]}{[{\bf x},{\bf x}][{{\bf x}^{\prime}},{{\bf
x}^{\prime}}]}\right)=R\cosh^{-1}\left(\frac{1}{R^{2}}[{\bf x},{{\bf
x}^{\prime}}]\right),$ (3)

where the inverse hyperbolic cosine with argument $x\in(1,\infty)$ is given by
(see (4.37.19) in Olver et al. (2010) [25])
$\cosh^{-1}x=\log\left(x+\sqrt{x^{2}-1}\right).$ Geodesics on
${\mathbf{H}}_{R}^{d}$ are great hyperbolas (i.e. intersections of
${\mathbf{H}}_{R}^{d}$ with planes through the origin) with constant speed
parametrizations (see p. 84 in Lee (1997) [20]). We also define a global
function $\rho:{\mathbf{H}}^{d}\times{\mathbf{H}}^{d}\to[0,\infty)$ which
represents the projection of global geodesic distance function (3) on
${\mathbf{H}}_{R}^{d}$ onto the corresponding unit radius hyperboloid
${\mathbf{H}}^{d}$, namely

$\rho({\widehat{\bf x}},{\widehat{\bf x}}^{\prime}):=d({\bf x},{{\bf
x}^{\prime}})/R,$ (4)

where ${\widehat{\bf x}}={\bf x}/R$ and ${\widehat{\bf x}}^{\prime}={{\bf
x}^{\prime}}/R$. Note that when we refer to $d({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime})$ below, we specifically mean that projected distance given by
(4).

###  2.2 Laplace’s equation and harmonics on the hyperboloid

Parametrizations of a submanifold embedded in either a Euclidean or Minkowski
space is given in terms of coordinate systems whose coordinates are
curvilinear. These are coordinates based on some transformation that converts
the standard Cartesian coordinates in the ambient space to a coordinate system
with the same number of coordinates as the dimension of the submanifold in
which the coordinate lines are curved.

The Laplace-Beltrami operator (Laplacian) in curvilinear coordinates
${\mathbf{\xi}}=(\xi^{1},\ldots,\xi^{d})$ on a $d$-dimensional Riemannian
manifold (a manifold together with a Riemannian metric $g$) is given by

$\Delta=\sum_{i,j=1}^{d}\frac{1}{\sqrt{|g|}}\frac{\partial}{\partial\xi^{i}}\left(\sqrt{|g|}g^{ij}\frac{\partial}{\partial\xi^{j}}\right),$
(5)

where $|g|=|\det(g_{ij})|,$ the infinitesimal distance is given by

$ds^{2}=\sum_{i,j=1}^{d}g_{ij}d\xi^{i}d\xi^{j},\ $ (6)

and

$\sum_{i=1}^{d}g_{ki}g^{ij}=\delta_{k}^{j},$

where $\delta_{i}^{j}\in\\{0,1\\}$ with $i,j\in{\mathbf{Z}}$, is the Kronecker
delta defined such that

$\delta_{i}^{j}:=\left\\{\begin{array}[]{ll}\displaystyle 1&\qquad\mathrm{if}\
i=j,\\\\[2.84544pt] \displaystyle 0&\qquad\mathrm{if}\ i\neq
j.\end{array}\right.$

For a submanifold, the relation between the metric tensor in the ambient space
and $g_{ij}$ of (5) and (6) is

$g_{ij}({\mathbf{\xi}})=\sum_{k,l=0}^{d}G_{kl}\frac{\partial
x^{k}}{\partial\xi^{i}}\frac{\partial x^{l}}{\partial\xi^{j}}.$

The ambient space for the hyperboloid is Minkowski, and therefore
$G_{ij}=\mathrm{diag}(1,-1,\ldots,-1)$.

The set of all geodesic polar coordinate systems corresponds to the many ways
one can put coordinates on a hyperbolic hypersphere, i.e., the Riemannian
submanifold $U\subset{\mathbf{H}}_{R}^{d}$ defined for a fixed ${{\bf
x}^{\prime}}\in{\mathbf{H}}_{R}^{d}$ such that $d({\bf x},{{\bf
x}^{\prime}})=b=const,$ where $b\in(0,\infty)$. These are coordinate systems
which correspond to subgroup chains starting with $O(d,1)\supset
O(d)\supset\cdots$, with standard geodesic polar coordinates given by (1)
being only one of them. (For a thorough description of these see section X.5
in Vilenkin (1968) [29].) They all share the property that they are described
by $(d+1)$-variables: $r\in[0,\infty)$ plus $d$-angles each being given by the
values $[0,2\pi)$, $[0,\pi]$, $[-\pi/2,\pi/2]$ or $[0,\pi/2]$ (see Izmest’ev
et al. (1999, 2001) [17, 18]).

In any of the geodesic polar coordinate systems, the geodesic distance between
two points on the submanifold is given by (cf. (3))

$d({\bf x},{{\bf x}^{\prime}})=R\cosh^{-1}\bigl{(}\cosh r\cosh
r^{\prime}-\sinh r\sinh r^{\prime}\cos\gamma\bigr{)},$ (7)

where $\gamma$ is the unique separation angle given in each hyperspherical
coordinate system. For instance, the separation angle in standard geodesic
polar coordinates (1) is given by the formula

$\displaystyle\cos\,\gamma=\cos(\phi-\phi^{\prime})\prod_{i=1}^{d-2}\sin\theta_{i}{\sin\theta_{i}}^{\prime}+\sum_{i=1}^{d-2}\cos\theta_{i}{\cos\theta_{i}}^{\prime}\prod_{j=1}^{i-1}\sin\theta_{j}{\sin\theta_{j}}^{\prime}.$
(8)

Corresponding separation angle formulae for any geodesic polar coordinate
system can be computed using (2), (3), and the associated formulae for the
appropriate inner-products.

The infinitesimal distance in a geodesic polar coordinate system on this
submanifold is

$ds^{2}=R^{2}(dr^{2}+\sinh^{2}r\ d\gamma^{2}),$ (9)

where an appropriate expression for $\gamma$ in a curvilinear coordinate
system is given. If one combines (1), (5), (8) and (9), then in a particular
geodesic polar coordinate system, Laplace’s equation on ${\mathbf{H}}_{R}^{d}$
is

$\Delta f=\frac{1}{R^{2}}\left[\frac{\partial^{2}f}{\partial r^{2}}+(d-1)\coth
r\frac{\partial f}{\partial
r}+\frac{1}{\sinh^{2}r}\Delta_{{\mathbf{S}}^{d-1}}f\right]=0,$ (10)

where $\Delta_{{\mathbf{S}}^{d-1}}$ is the corresponding Laplace-Beltrami
operator on ${\mathbf{S}}^{d-1}$ with unit radius.

From this point onwards, ${\mathbf{S}}^{d-1}$ will always refer to the
$(d-1)$-dimensional unit hypersphere, which is a compact Riemannian
submanifold with positive constant sectional curvature, embedded in
${\mathbf{R}}^{d}$ and given by the variety $x_{1}^{2}+\ldots+x_{d}^{2}=1$.

Geodesic polar coordinate systems partition ${\mathbf{H}}_{R}^{d}$ into a
family of concentric $(d-1)$-dimensional hyperspheres, each with a radius
$r\in(0,\infty),$ on which all possible hyperspherical coordinate systems for
${\mathbf{S}}^{d-1}$ may be used (see for instance, in Vilenkin (1968) [29]).
One then must also consider the limiting case for $r=0$ to fill out all of
${\mathbf{H}}_{R}^{d}$. In standard geodesic polar coordinates one can compute
the normalized hyperspherical harmonics in this space by solving the Laplace
equation using separation of variables which results in a general procedure
which is given explicitly in Izmest’ev et al. (1999, 2001) [17, 18]. These
angular harmonics are given as general expressions involving trigonometric
functions, Gegenbauer polynomials and Jacobi polynomials.

The harmonics in geodesic polar coordinate systems are given in terms of a
radial solution multiplied by the angular harmonics. The angular harmonics are
eigenfunctions of the Laplace-Beltrami operator on ${\mathbf{S}}^{d-1}$ with
unit radius which satisfy the following eigenvalue problem

$\Delta_{{\mathbf{S}}^{d-1}}Y_{l}^{K}({\widehat{\bf
x}})=-l(l+d-2)Y_{l}^{K}({\widehat{\bf x}}),$ (11)

where ${\widehat{\bf x}}\in{\mathbf{S}}^{d-1}$, $Y_{l}^{K}({\widehat{\bf x}})$
are normalized hyperspherical harmonics, $l\in{\mathbf{N}}_{0}$ is the angular
momentum quantum number, and $K$ stands for the set of $(d-2)$-quantum numbers
identifying degenerate harmonics for each $l$. The degeneracy as a function of
the dimension $d$ tells you how many linearly independent solutions exist for
a particular $l$ value. The hyperspherical harmonics are normalized such that

$\int_{{\mathbf{S}}^{d-1}}Y_{l}^{K}({\widehat{\bf
x}})\overline{Y_{l^{\prime}}^{K^{\prime}}({\widehat{\bf
x}})}d\omega=\delta_{l}^{l^{\prime}}\delta_{K}^{K^{\prime}},$

where $d\omega$ is a volume measure on ${\mathbf{S}}^{d-1}$ which is invariant
under the isometry group $SO(d)$ (cf. (13)), and for $x+iy=z\in{\mathbf{C}}$,
$\overline{z}=x-iy$, represents complex conjugation. The generalized Kronecker
delta $\delta_{K}^{K^{\prime}}$ (cf. (2.2)) is defined such that it equals 1
if all of the $(d-2)$-quantum numbers identifying degenerate harmonics for
each $l$ coincide, and equals zero otherwise.

Since the angular solutions (hyperspherical harmonics) are well-known (see
Chapter IX in Vilenkin (1968) [29]; Chapter 11 in Erdélyi et al. (1981) [12]),
we will now focus on the radial solutions, which satisfy the following
ordinary differential equation

$\frac{d^{2}u}{dr^{2}}+(d-1)\coth
r\frac{du}{dr}-\frac{l(l+d-2)}{\sinh^{2}r}u=0.$

Four solutions to this ordinary differential equation
$u_{1\pm}^{d,l},u_{2\pm}^{d,l}:(1\infty)\to{\mathbf{C}}$ are given by

${\displaystyle u_{1\pm}^{d,l}(\cosh
r)=\frac{1}{\sinh^{d/2-1}r}P_{d/2-1}^{\pm(d/2-1+l)}(\cosh r)},$

and

${\displaystyle u_{2\pm}^{d,l}(\cosh
r)=\frac{1}{\sinh^{d/2-1}r}Q_{d/2-1}^{\pm(d/2-1+l)}(\cosh r)},$

where $P_{\nu}^{\mu},Q_{\nu}^{\mu}:(1,\infty)\to{\mathbf{C}}$ are associated
Legendre functions of the first and second kind respectively (see for instance
Chapter 14 in Olver et al. (2010) [25]).

###  2.3 Fundamental solution of Laplace’s equation on the hyperboloid

Due to the fact that the space ${\mathbf{H}}_{R}^{d}$ is homogeneous with
respect to its isometry group, the pseudo-orthogonal group $SO(d,1)$, and
therefore an isotropic manifold, we expect that there exist a fundamental
solution on this space with spherically symmetric dependence. We specifically
expect these solutions to be given in terms of associated Legendre functions
of the second kind with argument given by $\cosh r$. This associated Legendre
function naturally fits our requirements because it is singular at $r=0$ and
vanishes at infinity, whereas the associated Legendre functions of the first
kind, with the same argument, are regular at $r=0$ and singular at infinity.

In computing a fundamental solution of the Laplacian on
${\mathbf{H}}_{R}^{d}$, we know that

$-\Delta{\mathcal{H}}_{R}^{d}({\bf x},{{\bf x}^{\prime}})=\delta_{g}({\bf
x},{{\bf x}^{\prime}}),$ (12)

where $g$ is the Riemannian metric on ${\mathbf{H}}_{R}^{d}$ and
$\delta_{g}({\bf x},{{\bf x}^{\prime}})$ is the Dirac delta function on the
manifold ${\mathbf{H}}_{R}^{d}$. The Dirac delta function is defined for an
open set $U\subset{\mathbf{H}}_{R}^{d}$ with ${\bf x},{{\bf
x}^{\prime}}\in{\mathbf{H}}_{R}^{d}$ such that

$\int_{U}\delta_{g}({\bf x},{{\bf
x}^{\prime}})d\mbox{vol}_{g}=\left\\{\begin{array}[]{ll}\displaystyle
1&\qquad\mathrm{if}\ {{\bf x}^{\prime}}\in U,\\\\[2.84544pt] \displaystyle
0&\qquad\mathrm{if}\ {{\bf x}^{\prime}}\notin U,\end{array}\right.$

where $d\mbox{vol}_{g}$ is a volume measure, invariant under the isometry
group $SO(d,1)$ of the Riemannian manifold ${\mathbf{H}}_{R}^{d}$, given (in
standard geodesic polar coordinates) by

$d\mbox{vol}_{g}=R^{d}\sinh^{d-1}rd\omega:=R^{d}\sinh^{d-1}r\sin^{d-2}\theta_{d-1}\cdots\sin\theta_{2}d\theta_{1}\cdots
d\theta_{d-1}.$ (13)

Notice that as $r\to 0^{+}$ that $d\mbox{vol}_{g}$ goes to the Euclidean
measure, invariant under the Euclidean motion group $E(d)$, in spherical
coordinates. Therefore in spherical coordinates, we have the following

$\delta_{g}({\bf x},{{\bf
x}^{\prime}})=\frac{\delta(r-r^{\prime})}{R^{d}\sinh^{d-1}r^{\prime}}\frac{\delta(\theta_{1}-\theta_{1}^{\prime})\cdots\delta(\theta_{d-1}-\theta_{d-1}^{\prime})}{\sin\theta_{2}^{\prime}\cdots\sin^{d-2}\theta_{d-1}^{\prime}}.$
(14)

In general since we can add any harmonic function to a fundamental solution
for the Laplacian and still have a fundamental solution, we will use this
freedom to make our fundamental solution as simple as possible. It is
reasonable to expect that there exists a particular spherically symmetric
fundamental solution ${\mathcal{H}}_{R}^{d}({\bf x},{{\bf x}^{\prime}})$ on
the hyperboloid with pure radial $\rho({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime})=d({\bf x},{{\bf x}^{\prime}})/R$ (cf. (4)) and constant angular
dependence (invariant under rotations centered about the origin), due to the
influence of the point-like nature of the Dirac delta function. For a
spherically symmetric solution to the Laplace equation, the corresponding
$\Delta_{{\mathbf{S}}^{d-1}}$ term vanishes since only the $l=0$ term
survives. In other words, we expect there to exist a fundamental solution of
Laplace’s equation such that ${\mathcal{H}}_{R}^{d}({\bf x},{{\bf
x}^{\prime}})=f(\rho)$.

In Cohl & Kalnins (2011) [8], we have proven that on the $R$-radius
hyperboloid ${\mathbf{H}}_{R}^{d}$, a fundamental solution of Laplace’s
equation can be given as follows.

######  Theorem 2.1.

Let $d\in\\{2,3,\ldots\\}.$ Define
${\mathcal{I}}_{d}:(0,\infty)\to{\mathbf{R}}$ as

${\mathcal{I}}_{d}(\rho):=\int_{\rho}^{\infty}\frac{dx}{\sinh^{d-1}x},$

${\bf x},{{\bf x}^{\prime}}\in{\mathbf{H}}_{R}^{d}$, and
${\mathcal{H}}_{R}^{d}:({\mathbf{H}}_{R}^{d}\times{\mathbf{H}}_{R}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbf{H}}_{R}^{d}\\}\to{\mathbf{R}}$ defined such
that

${\mathcal{H}}_{R}^{d}({\bf x},{\bf
x}^{\prime}):={\displaystyle\frac{\Gamma\left(d/2\right)}{2\pi^{d/2}R^{d-2}}{\mathcal{I}}_{d}(\rho)},$

where $\rho:=\cosh^{-1}\left([{\widehat{\bf x}},{\widehat{\bf
x}}^{\prime}]\right)$ is the geodesic distance between ${\widehat{\bf x}}$ and
${\widehat{\bf x}}^{\prime}$ on the pseudo-sphere of unit radius
${\mathbf{H}}^{d}$, with ${\widehat{\bf x}}={\bf x}/R,$ ${\widehat{\bf
x}}^{\prime}={{\bf x}^{\prime}}/R$, then ${\mathcal{H}}_{R}^{d}$ is a
fundamental solution for $-\Delta$ where $\Delta$ is the Laplace-Beltrami
operator on ${\mathbf{H}}_{R}^{d}$. Moreover,

$\displaystyle{\mathcal{I}}_{d}(\rho)=\left\\{\begin{array}[]{ll}\displaystyle(-1)^{d/2-1}\frac{(d-3)!!}{(d-2)!!}\Biggl{[}\log\coth\frac{\rho}{2}+\cosh\rho\sum_{k=1}^{d/2-1}\frac{(2k-2)!!(-1)^{k}}{(2k-1)!!\sinh^{2k}\rho}\Biggr{]}&\mathrm{if}\
d\ \mathrm{even},\\\\[17.07182pt]
\left\\{\begin{array}[]{l}\displaystyle(-1)^{(d-1)/2}\Biggl{[}\frac{(d-3)!!}{(d-2)!!}\\\\[11.38092pt]
\displaystyle\hskip
34.14322pt+\left(\frac{d-3}{2}\right)!\sum_{k=1}^{(d-1)/2}\frac{(-1)^{k}\coth^{2k-1}\rho}{(2k-1)(k-1)!((d-2k-1)/2)!}\Biggr{]},\\\\[12.80365pt]
\mathrm{or}\\\\[0.0pt]
\displaystyle(-1)^{(d-1)/2}\frac{(d-3)!!}{(d-2)!!}\left[1+\cosh\rho\sum_{k=1}^{(d-1)/2}\frac{(2k-3)!!(-1)^{k}}{(2k-2)!!\sinh^{2k-1}\rho}\right],\end{array}\right\\}&\mathrm{if}\
d\ \mathrm{odd}.\end{array}\right.$
$\displaystyle=\frac{1}{(d-1)\cosh^{d-1}\rho}\,{}_{2}F_{1}\left(\frac{d-1}{2},\frac{d}{2};\frac{d+1}{2};\frac{1}{\cosh^{2}\rho}\right),$
$\displaystyle=\frac{1}{(d-1)\cosh\rho\,\sinh^{d-2}\rho}\,{}_{2}F_{1}\left(\frac{1}{2},1;\frac{d+1}{2};\frac{1}{\cosh^{2}\rho}\right),$
$\displaystyle=\frac{e^{-i\pi(d/2-1)}}{2^{d/2-1}\Gamma\left(d/2\right)\sinh^{d/2-1}\rho}\,Q_{d/2-1}^{d/2-1}(\cosh\rho),$

where $!!$ is the double factorial, ${}_{2}F_{1}$ is the Gauss hypergeometric
function, and $Q_{\nu}^{\mu}$ is the associated Legendre function of the
second kind.

For a proof of this theorem, see Cohl & Kalnins (2011) [8].

##  3 Fourier expansions for a Green’s function on the hyperboloid

Now we compute the Fourier expansions for a fundamental solution of the
Laplace-Beltrami operator on ${\mathbf{H}}_{R}^{d}$.

###  3.1 Fourier expansion for a fundamental solution of the Laplacian on
${\mathbf{H}}_{R}^{2}$

The generating function for Chebyshev polynomials of the first kind (Fox &
Parker (1968) [14], p. 51) is given as

$\frac{1-z^{2}}{1+z^{2}-2xz}=\sum_{n=0}^{\infty}\epsilon_{n}T_{n}(x)z^{n},$
(17)

where $|z|<1,$ $T_{n}:[-1,1]\to{\mathbf{R}}$ is the Chebyshev polynomial of
the first kind defined as $T_{l}(x):=\cos(l\cos^{-1}x),$ and
$\epsilon_{n}:=2-\delta_{n}^{0}$ is the Neumann factor (see p. 744 in Morse &
Feshbach (1953) [22]), commonly-occurring in Fourier cosine series. If
substitute $z=e^{-\eta}$ with $\eta\in(0,\infty)$ in (17), then we obtain

$\frac{\sinh\eta}{\cosh\eta-\cos\psi}=\sum_{n=0}^{\infty}\epsilon_{n}\cos(n\psi)e^{-n\eta}.$
(18)

Integrating both sides of (18) with respect to $\eta$, we obtain the following
formula (cf. Magnus, Oberhettinger & Soni (1966) [21], p. 259)

$\log\left(1+z^{2}-2z\cos\psi\right)=-2\sum_{n=1}^{\infty}\frac{\cos(n\psi)}{n}z^{n}.$
(19)

In Euclidean space ${\mathbf{R}}^{d}$, a Green’s function for Laplace’s
equation (fundamental solution for the Laplacian) is well-known and is given
in the following theorem (see Folland (1976) [13]; p. 94, Gilbarg & Trudinger
(1983) [15]; p. 17, Bers et al. (1964) [4], p. 211).

######  Theorem 3.1.

Let $d\in{\mathbf{N}}$. Define

${\mathcal{G}}^{d}({\bf x},{\bf
x}^{\prime})=\left\\{\begin{array}[]{ll}\displaystyle\frac{\Gamma(d/2)}{2\pi^{d/2}(d-2)}\|{\bf
x}-{\bf x}^{\prime}\|^{2-d}&\qquad\mathrm{if}\ d=1\mathrm{\ or\ }d\geq
3,\\\\[10.0pt] \displaystyle\frac{1}{2\pi}\log\|{\bf x}-{\bf
x}^{\prime}\|^{-1}&\qquad\mathrm{if}\ d=2,\end{array}\right.$

then ${\mathcal{G}}^{d}$ is a fundamental solution for $-\Delta$ in Euclidean
space ${\mathbf{R}}^{d}$, where $\Delta$ is the Laplace operator in
${\mathbf{R}}^{d}$.

Therefore if we take $z=r_{<}/r_{>}$ in (19), where
$r_{\lessgtr}:={\min\atop\max}\\{r,r^{\prime}\\}$ with
$r,r^{\prime}\in[0,\infty),$ then using polar coordinates, we can derive the
Fourier expansion for a fundamental solution of the Laplacian in Euclidean
space for $d=2$ (cf. Theorem 3.1), namely

${\mathfrak{g}}^{2}:=\log\|{\bf x}-{{\bf x}^{\prime}}\|=\log
r_{>}-\sum_{n=1}^{\infty}\frac{\cos(n(\phi-\phi^{\prime}))}{n}\biggl{(}\frac{r_{<}}{r_{>}}\biggr{)}^{n},$
(20)

where ${\mathfrak{g}}^{2}=-2\pi{\mathcal{G}}^{2}$ (cf. Theorem 3.1). On the
hyperboloid for $d=2$ we have a fundamental solution of Laplace’s equation
given by

${\mathfrak{h}}^{2}:=\log\coth\frac{1}{2}d({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime})=\frac{1}{2}\log\frac{\cosh d({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime})+1}{\cosh d({\widehat{\bf x}},{\widehat{\bf x}}^{\prime})-1},$

where ${\mathfrak{h}}^{2}=2\pi{\mathcal{H}}_{R}^{2}$ (cf. Theorem 2.1 and (32)
below). Note that because of the $R^{d-2}$ dependence of a fundamental
solution of Laplace’s equation for $d=2$ in Theorem 2.1, there is no strict
dependence on $R$ for ${\mathcal{H}}_{R}^{2}$ or ${\mathfrak{h}}^{2}$, but
will retain the notation nonetheless. In standard geodesic polar coordinates
on ${\mathbf{H}}_{R}^{2}$ (cf. (1)), using (7) and
$\cos\gamma=\cos(\phi-\phi^{\prime})$ (cf. (8)) produces

$\cosh d({\widehat{\bf x}},{\widehat{\bf x}}^{\prime})=\cosh r\cosh
r^{\prime}-\sinh r\sinh r^{\prime}\cos(\phi-\phi^{\prime}),$

therefore

${\mathfrak{h}}^{2}=\frac{1}{2}\log\frac{\cosh r\cosh r^{\prime}+1-\sinh
r\sinh r^{\prime}\cos(\phi-\phi^{\prime})}{\cosh r\cosh r^{\prime}-1-\sinh
r\sinh r^{\prime}\cos(\phi-\phi^{\prime})}.$

Replacing $\psi=\phi-\phi^{\prime}$ and rearranging the logarithms yield

${\mathfrak{h}}^{2}=\frac{1}{2}\log\frac{\cosh r\cosh r^{\prime}+1}{\cosh
r\cosh
r^{\prime}-1}+\frac{1}{2}\log\left(1-z_{+}\cos\psi\right)-\frac{1}{2}\log\left(1-z_{-}\cos\psi\right),$

where

$z_{\pm}:=\frac{\sinh r\sinh r^{\prime}}{\cosh r\cosh r^{\prime}\pm 1}.$

Note that $z_{\pm}\in(0,1)$ for $r,r^{\prime}\in(0,\infty)$. We have the
following MacLaurin series

$\log(1-x)=-\sum_{n=1}^{\infty}\frac{x^{n}}{n},$

where $x\in[-1,1)$. Therefore away from the singularity at ${\bf x}={{\bf
x}^{\prime}}$ we have

$\lambda_{\pm}:=\log\left(1-z_{\pm}\cos\psi\right)=-\sum_{k=1}^{\infty}\frac{z_{\pm}^{k}}{k}\cos^{k}\psi.$
(21)

We can expand the powers of cosine using the following trigonometric identity

$\cos^{k}\psi=\frac{1}{2^{k}}\sum_{n=0}^{k}\left(\begin{array}[]{c}\displaystyle\\!\\!k\\\\[1.0pt]
\displaystyle\\!\\!n\end{array}\\!\\!\right)\cos[(2n-k)\psi],$

which is the standard expansion for powers using Chebyshev polynomials (see
for instance p. 52 in Fox & Parker (1968) [14]). Inserting this expression in
(21), we obtain the following double-summation expression

$\lambda_{\pm}=-\sum_{k=1}^{\infty}\sum_{n=0}^{k}\frac{z_{\pm}^{k}}{2^{k}k}\left(\begin{array}[]{c}\displaystyle\\!\\!k\\\\[1.0pt]
\displaystyle\\!\\!n\end{array}\\!\\!\right)\cos[(2n-k)\psi].$ (22)

Now we perform a double-index replacement in (22). We break this sum into two
separate sums, one for $k\leq 2n$ and another for $k\geq 2n$. There is an
overlap when both sums satisfy the equality, and in that situation we must
halve after we sum over both sums. If $k\leq 2n$, make the substitution
$k^{\prime}=k-n$ and $n^{\prime}=2n-k$. It follows that
$k=2k^{\prime}+n^{\prime}$ and $n=n^{\prime}+k^{\prime}$, therefore

$\left(\begin{array}[]{c}\displaystyle\\!\\!k\\\\[1.0pt]
\displaystyle\\!\\!n\end{array}\\!\\!\right)=\left(\begin{array}[]{c}\displaystyle\\!\\!2k^{\prime}+n^{\prime}\\\\[1.0pt]
\displaystyle\\!\\!n^{\prime}+k^{\prime}\end{array}\\!\\!\right)=\left(\begin{array}[]{c}\displaystyle\\!\\!2k^{\prime}+n^{\prime}\\\\[1.0pt]
\displaystyle\\!\\!n^{\prime}+k^{\prime}\end{array}\\!\\!\right).$

If $k\geq 2n$ make the substitution $k^{\prime}=n$ and $n^{\prime}=k-2n$. Then
$k=2k^{\prime}+n^{\prime}$ and $n=k^{\prime}$, therefore

$\left(\begin{array}[]{c}\displaystyle\\!\\!k\\\\[1.0pt]
\displaystyle\\!\\!n\end{array}\\!\\!\right)=\left(\begin{array}[]{c}\displaystyle\\!\\!2k^{\prime}+n^{\prime}\\\\[1.0pt]
\displaystyle\\!\\!n\end{array}\\!\\!\right)=\left(\begin{array}[]{c}\displaystyle\\!\\!2k^{\prime}+n^{\prime}\\\\[1.0pt]
\displaystyle\\!\\!k^{\prime}+n^{\prime}\end{array}\\!\\!\right),$

where the equalities of the binomial coefficients are confirmed using the
following identity

$\left(\begin{array}[]{c}\displaystyle\\!\\!n\\\\[1.0pt]
\displaystyle\\!\\!k\end{array}\\!\\!\right)=\left(\begin{array}[]{c}\displaystyle\\!\\!n\\\\[1.0pt]
\displaystyle\\!\\!n-k\end{array}\\!\\!\right),$

where $n,k\in{\mathbf{Z}}$, except where $k<0$ or $n-k<0$. To take into
account the double-counting which occurs when $k=2n$ (which occurs when
$n^{\prime}=0$), we introduce a factor of $\epsilon_{n^{\prime}}/2$ into the
expression (and relabel $k^{\prime}\mapsto k$ and $n^{\prime}\mapsto n$). We
are left with

$\lambda_{\pm}=-\frac{1}{2}\sum_{k=1}^{\infty}\frac{z_{\pm}^{2k}}{2^{k}k}\left(\begin{array}[]{c}\displaystyle\\!\\!2k\\\\[1.0pt]
\displaystyle\\!\\!k\end{array}\\!\\!\right)-2\sum_{n=1}^{\infty}\cos(n\psi)\sum_{k=0}^{\infty}\frac{z_{\pm}^{2k+n}}{2^{2k+n}(2k+n)}\left(\begin{array}[]{c}\displaystyle\\!\\!2k+n\\\\[1.0pt]
\displaystyle\\!\\!k\end{array}\\!\\!\right).$ (23)

If we substitute

$\left(\begin{array}[]{c}\displaystyle\\!\\!2k\\\\[1.0pt]
\displaystyle\\!\\!k\end{array}\\!\\!\right)=\frac{2^{2k}\left(\frac{1}{2}\right)_{k}}{k!}$

into the first term of (23), then we obtain

$I_{\pm}:=-\frac{1}{2}\sum_{k=1}^{\infty}\frac{\left(\frac{1}{2}\right)_{k}z_{\pm}^{2k}}{k!k}=-\int_{0}^{z_{\pm}}\frac{dz_{\pm}^{\prime}}{z_{\pm}^{\prime}}\sum_{k=1}^{\infty}\frac{\left(\frac{1}{2}\right)_{k}{z_{\pm}^{\prime}}^{2k}}{k!}=-\int_{0}^{z_{\pm}}\frac{d{z_{\pm}^{\prime}}}{z_{\pm}^{\prime}}\left[\frac{1}{\sqrt{1-{z_{\pm}^{\prime}}^{2}}}-1\right].$

We are left with

$I_{\pm}=-\log 2+\log\left(1+\sqrt{1-z_{\pm}^{2}}\right)=-\log
2+\log\left(\frac{(\cosh r_{>}\pm 1)(\cosh r_{<}+1)}{\cosh r\cosh
r^{\prime}\pm 1}\right).$

If we substitute

$\displaystyle\left(\begin{array}[]{c}\displaystyle\\!\\!2k+n\\\\[1.0pt]
\displaystyle\\!\\!k\end{array}\\!\\!\right)=\frac{\displaystyle
2^{2k}\left(\frac{n+1}{2}\right)_{k}\left(\frac{n+2}{2}\right)_{k}}{k!(n+1)_{k}},$

into the second term of (23), then the Fourier coefficient reduces to

$\displaystyle J_{\pm}$ $\displaystyle:=$
$\displaystyle\frac{1}{2^{n-1}}\sum_{k=0}^{\infty}\frac{\displaystyle\left(\frac{n+1}{2}\right)_{k}\left(\frac{n+2}{2}\right)_{k}}{\displaystyle
k!(n+1)_{k}}\frac{z_{\pm}^{2k+n}}{2k+n}$ $\displaystyle=$
$\displaystyle\frac{1}{2^{n-1}}\int_{0}^{z_{\pm}}dz_{\pm}^{\prime}{z_{\pm}^{\prime}}^{n-1}\sum_{k=0}^{\infty}\frac{\displaystyle\left(\frac{n+1}{2}\right)_{k}\left(\frac{n+2}{2}\right)_{k}}{k!(n+1)_{k}}{z_{\pm}^{\prime}}^{2k}.$

The series in the integrand is a Gauss hypergeometric function which can be
given as

$\sum_{k=0}^{\infty}\frac{\displaystyle\left(\frac{n+1}{2}\right)_{k}\left(\frac{n+2}{2}\right)_{k}}{k!(n+1)_{k}}z^{2k}=\frac{2^{n}n!}{z^{n}\sqrt{1-z^{2}}}P_{0}^{-n}\left(\sqrt{1-z^{2}}\right),$

where $P_{0}^{-n}$ is an associated Legendre function of the first kind with
vanishing degree and order given by $-n$. This is a consequence of

${}_{2}F_{1}\left(a,b;a+b-\frac{1}{2};x\right)=2^{2+b-3/2}\Gamma\left(a+b-\frac{1}{2}\right)\frac{x^{(3-2a-2b)/4}}{\sqrt{1-x}}P_{b-a-1/2}^{3/2-a-b}\left(\sqrt{1-x}\right),$

where $x\in(0,1)$ (see for instance Magnus, Oberhettinger & Soni (1966) [21],
p. 53), and the Legendre function is evaluated using (cf. (8.1.2) in
Abramowitz & Stegun (1972) [1])

$P_{0}^{-n}(x)=\frac{1}{n!}\left(\frac{1-x}{1+x}\right)^{n/2},$

where $n\in{\mathbf{N}}_{0}$. Therefore the Fourier coefficient is given by

$J_{\pm}=2\int_{\sqrt{1-z_{\pm}^{2}}}^{1}\frac{dz_{\pm}^{\prime}}{1-{z_{\pm}^{\prime}}^{2}}\left(\frac{1-z_{\pm}^{\prime}}{1+z_{\pm}^{\prime}}\right)^{n/2}=\frac{2}{n}\left[\frac{1-\sqrt{1-z_{\pm}^{2}}}{1+\sqrt{1-z_{\pm}^{2}}}\right]^{n/2}.$

Finally we have

$\displaystyle\lambda_{\pm}=-\log 2+\log\left(\frac{(\cosh r_{>}\pm 1)(\cosh
r_{<}+1)}{\cosh r\cosh r^{\prime}\pm 1}\right)$ $\displaystyle\hskip
36.98866pt-2\sum_{n=1}^{\infty}\frac{\cos(n\psi)}{n}\left[\frac{(\cosh
r_{>}\mp 1)(\cosh r_{<}-1)}{(\cosh r_{>}\pm 1)(\cosh r_{<}+1)}\right]^{n/2},$

and the Fourier expansion for a fundamental solution of Laplace’s equation for
the $d=2$ hyperboloid is given by

$\displaystyle{\mathfrak{h}}^{2}=\frac{1}{2}\log\frac{\cosh r_{>}+1}{\cosh
r_{>}-1}$
$\displaystyle{}\displaystyle+\sum_{n=1}^{\infty}\frac{\cos(n(\phi-\phi^{\prime}))}{n}\left[\frac{\cosh
r_{<}-1}{\cosh r_{<}+1}\right]^{n/2}\left\\{\left[\frac{\cosh r_{>}+1}{\cosh
r_{>}-1}\right]^{n/2}-\left[\frac{\cosh r_{>}-1}{\cosh
r_{>}+1}\right]^{n/2}\right\\}.$ (24)

This exactly matches up to the Euclidean Fourier expansion
${\mathfrak{g}}^{2}$ (20) as $r,r^{\prime}\to 0^{+}$.

###  3.2 Fourier expansion for a fundamental solution of the Laplacian on
${\mathbf{H}}_{R}^{3}$

The Fourier expansion for a fundamental solution of the Laplacian in three-
dimensional Euclidean space (here given in standard spherical coordinates
${\bf x}=(r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta)$) is given by
(cf. Theorem 3.1, and see (1.3) in Cohl et al. (2001) [9])

$\displaystyle{\mathcal{G}}^{3}\simeq{\mathfrak{g}}^{3}:=\frac{1}{\|{\bf
x}-{{\bf x}^{\prime}}\|}$
$\displaystyle=\frac{1}{\pi\sqrt{rr^{\prime}\sin\theta\sin\theta^{\prime}}}\sum_{m=-\infty}^{\infty}e^{im(\phi-\phi^{\prime})}Q_{m-1/2}\left(\frac{r^{2}+{r^{\prime}}^{2}-2rr^{\prime}\cos\theta\cos\theta^{\prime}}{2rr^{\prime}\sin\theta\sin\theta^{\prime}}\right).$

These associated Legendre functions, toroidal harmonics, are given in terms of
complete elliptic integrals of the first and second kind (cf. (22–26) in Cohl
& Tohline (1999) [10]). Since $Q_{-1/2}(z)$ is given through (cf. (8.13.3) in
Abramowitz & Stegun (1972) [1])

$Q_{-1/2}(z)=\sqrt{\frac{2}{z+1}}K\left(\sqrt{\frac{2}{z+1}}\right),$

the $m=0$ component for ${\mathfrak{g}}^{3}$ is given by

$\left.{\mathfrak{g}}^{3}\right|_{m=0}=\frac{2}{\pi\sqrt{r^{2}+{r^{\prime}}^{2}-2rr^{\prime}\cos(\theta+\theta^{\prime})}}K\left(\sqrt{\frac{4rr^{\prime}\sin\theta\sin\theta^{\prime}}{r^{2}+{r^{\prime}}^{2}-2rr^{\prime}\cos(\theta+\theta^{\prime})}}\right).$
(25)

A fundamental solution of the Laplacian in standard geodesic polar coordinates
on ${\mathbf{H}}_{R}^{3}$ is given by (cf. Theorem 2.1 and (32) below).

$\displaystyle\displaystyle{\mathfrak{h}}^{3}({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime}):=\coth d({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime})-1=\frac{\cosh d({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime})}{\sqrt{\cosh^{2}d({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime})-1}}-1$ $\displaystyle\hskip 93.89418pt=\frac{\cosh r\cosh
r^{\prime}-\sinh r\sinh r^{\prime}\cos\gamma}{\sqrt{(\cosh r\cosh
r^{\prime}-\sinh r\sinh r^{\prime}\cos\gamma)^{2}-1}}-1,$

where ${\mathfrak{h}}^{3}=4\pi R{\mathcal{H}}_{R}^{3},$ and ${\bf x},{{\bf
x}^{\prime}}\in{\mathbf{H}}_{R}^{3}$, such that ${\widehat{\bf x}}={\bf x}/R$
and ${\widehat{\bf x}}^{\prime}={{\bf x}^{\prime}}/R$. In standard geodesic
polar coordinates (cf. (8)) we have

$\cos\gamma=\cos\theta\cos\theta^{\prime}+\sin\theta\sin\theta^{\prime}\cos(\phi-\phi^{\prime}).$
(26)

Replacing $\psi=\phi-\phi^{\prime}$ and defining

$A:=\cosh r\cosh r^{\prime}-\sinh r\sinh
r^{\prime}\cos\theta\cos\theta^{\prime},$

and

$B:=\sinh r\sinh r^{\prime}\sin\theta\sin\theta^{\prime},$

we have in the standard manner, the Fourier coefficients ${\sf
H}_{m}^{1/2}:[0,\infty)^{2}\times[0,\pi]^{2}\to{\mathbf{R}}$ of the expansion
(cf. (33) below)

${\mathfrak{h}}^{3}({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime})=\sum_{m=0}^{\infty}\cos(m(\phi-\phi^{\prime})){\sf
H}_{m}^{1/2}(r,r^{\prime},\theta,\theta^{\prime}),$ (27)

defined by

${\displaystyle{\sf
H}_{m}^{1/2}(r,r^{\prime},\theta,\theta^{\prime}):=-\delta_{n}^{0}+\frac{\epsilon_{m}}{\pi}}{\displaystyle\int_{0}^{\pi}\frac{\left(A/B-\cos\psi\right)\cos(m\psi)d\psi}{\sqrt{\left(\cos\psi-\frac{A+1}{B}\right)\left(\cos\psi-\frac{A-1}{B}\right)}}}.$
(28)

If we make the substitution $x=\cos\psi$, this integral can be converted to

${\displaystyle{\sf
H}_{m}^{1/2}(r,r^{\prime},\theta,\theta^{\prime})=-\delta_{n}^{0}+\frac{\epsilon_{m}}{\pi}}{\displaystyle\int_{-1}^{1}\frac{\left(A/B-x\right)T_{m}(x)dx}{\sqrt{(1-x)(1+x)\left(x-\frac{A+1}{B}\right)\left(x-\frac{A-1}{B}\right)}}},$
(29)

where $T_{m}$ is the Chebyshev polynomial of the first kind. Since $T_{m}(x)$
is expressible as a finite sum over powers of $x$, (29) involves the square
root of a quartic multiplied by a rational function of $x$, which by
definition is an elliptic integral (see for instance Byrd & Friedman (1954)
[5]). We can directly compute (29) using Byrd & Friedman (1954) ([5],
(253.11)). If we define

$d:=-1,\ y:=-1,\ c:=1,\ b:=\frac{A-1}{B},\ a:=\frac{A+1}{B},$ (30)

(clearly $d\leq y<c<b<a$), then we can express the Fourier coefficient (29),
as a linear combination of integrals, each of the form (see Byrd & Friedman
(1954) [5], (253.11))

$\int_{y}^{c}\frac{x^{p}dx}{\sqrt{(a-x)(b-x)(c-x)(x-d)}}=c^{p}g\int_{0}^{u_{1}}\left[\frac{1-\alpha_{1}^{2}\mathrm{sn}^{2}u}{1-\alpha^{2}\mathrm{sn}^{2}u}\right]^{p}du,$
(31)

where $p\in\\{0,\ldots,m+1\\}$. In this expression $\mathrm{sn}$ is a Jacobi
elliptic function (see for instance Chapter 22 in Olver et al. (2010) [25]).

Byrd & Friedman (1954) [5] give a procedure for computing (31) for all
$m\in{\mathbf{N}}_{0}$. These integrals will be given in terms of complete
elliptic integrals of the first three kinds (see the discussion in Byrd &
Friedman (1954) [5], p. 201, 204, and p. 205). To this effect, we have the
following definitions from (253.11) in Byrd & Friedman (1954) [5], namely

$\alpha^{2}=\frac{c-d}{b-d}<1,$ $\alpha_{1}^{2}=\frac{b(c-d)}{c(b-d)},$
$g=\frac{2}{\sqrt{(a-c)(b-d)}},$
$\varphi=\sin^{-1}\sqrt{\frac{(b-d)(c-y)}{(c-d)(b-y)}},$ $u_{1}=F(\varphi,k),$
$k^{2}=\frac{(a-b)(c-d)}{(a-c)(b-d)},$

with $k^{2}<\alpha^{2}$. For our specific choices in (30), these reduce to

$\alpha^{2}=\frac{2B}{A+B-1},$ $\alpha_{1}^{2}=\frac{2(A-1)}{A+B-1},$
$g=\frac{2B}{\sqrt{(A+B-1)(A-B+1)}},$ $k^{2}=\frac{4B}{(A+B-1)(A-B+1)},$
$\varphi=\frac{\pi}{2},$

and

$u_{1}=K(k).$

Specific cases include

$\int_{y}^{c}\frac{dx}{\sqrt{(a-x)(b-x)(c-x)(x-d)}}=gK(k)$

(Byrd & Friedman (1954) [5], (340.00)) and

$\int_{y}^{c}\frac{xdx}{\sqrt{(a-x)(b-x)(c-x)(x-d)}}=\frac{cg}{\alpha^{2}}\left[\alpha_{1}^{2}K(k)+(\alpha^{2}-\alpha_{1}^{2})\Pi(\alpha,k)\right]$

(Byrd & Friedman (1954) [5], (340.01)).

In general we have

$\int_{y}^{c}\frac{x^{p}dx}{\sqrt{(a-x)(b-x)(c-x)(x-d)}}=\frac{c^{p}g\alpha_{1}^{2p}p!}{\alpha^{2p}}\sum_{j=0}^{p}\frac{(\alpha^{2}-\alpha_{1}^{2})^{j}}{\alpha_{1}^{2j}j!(p-j)!}V_{j}$

(Byrd & Friedman (1954) [5], (340.04)), where

$V_{0}=K(k),$ $V_{1}=\Pi(\alpha,k),$
$V_{2}=\frac{1}{2(\alpha^{2}-1)(k^{2}-\alpha^{2})}\left[(k^{2}-\alpha^{2})K(k)+\alpha^{2}E(k)+(2\alpha^{2}k^{2}+2\alpha^{2}-\alpha^{4}-3k^{2})\Pi(\alpha,k)\right],$

and larger values of $V_{j}$ can be computed using the following recurrence
relation

$\displaystyle V_{m+3}=\frac{1}{2(m+2)(1-\alpha^{2})(k^{2}-\alpha^{2})}$
$\displaystyle\hskip
51.21504pt\times\bigl{[}(2m+1)k^{2}V_{m}+2(m+1)(\alpha^{2}k^{2}+\alpha^{2}-3k^{2})V_{m+1}$
$\displaystyle\hskip
102.43008pt+(2m+3)(\alpha^{4}-2\alpha^{2}k^{2}-2\alpha^{2}+3k^{2})V_{m+2}\bigr{]}$

(see Byrd & Friedman (1954) [5], (336.00–03)). For instance,

$\eqalign{\int_{y}^{c}\frac{x^{2}dx}{\sqrt{(a-x)(b-x)(c-x)(x-d)}}\cr\hskip
28.45274pt=\frac{c^{2}g}{\alpha^{4}}\left[\alpha_{1}^{4}K(k)+2\alpha_{1}^{2}(\alpha^{2}-\alpha_{1}^{2})\Pi(\alpha,k)+(\alpha^{2}-\alpha_{1}^{2})^{2}V_{2}\right]}$

(see Byrd & Friedman (1954) [5], (340.02)).

In general, the Fourier coefficients for ${\mathfrak{h}}^{3}$ will be given in
terms of complete elliptic integrals of the first three kinds. Let’s directly
compute the $m=0$ component, in which (29) reduces to

${\displaystyle{\sf
H}_{0}^{1/2}(r,r^{\prime},\theta,\theta^{\prime})=-1+\frac{1}{\pi}}{\displaystyle\int_{-1}^{1}\frac{\left(A/B-x\right)dx}{\sqrt{(1-x)(1+x)\left(x-\frac{A+1}{B}\right)\left(x-\frac{A-1}{B}\right)}}}.$

Therefore using the above formulae, we have

$\displaystyle{\mathfrak{h}}_{3}|_{m=0}={\sf
H}_{0}^{1/2}(r,r^{\prime},\theta,\theta^{\prime})$
$\displaystyle=-1+\frac{2K(k)}{\pi\sqrt{(A-B+1)(A+B-1)}}+\frac{2(A-B-1)\Pi(\alpha,k)}{\pi\sqrt{(A-B+1)(A+B-1)}}$
$\displaystyle=-1+\frac{2}{\pi}\left\\{K(k)+\left[\cosh r\cosh
r^{\prime}-\sinh r\sinh
r^{\prime}\cos(\theta-\theta^{\prime})-1\right]\Pi(\alpha,k)\right\\}$
$\displaystyle\hskip 73.97733pt\times\left[\cosh r\cosh r^{\prime}-\sinh
r\sinh r^{\prime}\cos(\theta-\theta^{\prime})+1\right]^{-1/2}$
$\displaystyle\hskip 73.97733pt\times\left[\cosh r\cosh r^{\prime}-\sinh
r\sinh r^{\prime}\cos(\theta+\theta^{\prime})-1\right]^{-1/2}.$

Note that the Fourier coefficients

${\mathfrak{h}}^{3}|_{m=0}\to{\mathfrak{g}}^{3}|_{m=0},$

in the limit as $r,r^{\prime}\rightarrow 0^{+}$, where
${\mathfrak{g}}^{3}|_{m=0}$ is given in (25). This is expected since
${\mathbf{H}}_{R}^{3}$ is a manifold.

###  3.3 Fourier expansion for a fundamental solution of the Laplacian on
${\mathbf{H}}_{R}^{d}$

For the $d$-dimensional Riemannian manifold ${\mathbf{H}}_{R}^{d}$, with
$d\geq 2$, one can expand a fundamental solution of the Laplace-Beltrami
operator in an azimuthal Fourier series. One may Fourier expand, in terms of
the azimuthal coordinate, a fundamental solution of the Laplace-Beltrami
operator in any rotationally-invariant coordinate systems which admits
solutions via separation of variables. In Euclidean space, there exist non-
subgroup-type rotationally invariant coordinate systems which are separable
for Laplace’s equation. All separable coordinate systems for Laplace’s
equation in $d$-dimensional Euclidean space ${\mathbf{R}}^{d}$ are known. In
fact, this is also true for separable coordinate systems on
${\mathbf{H}}_{R}^{d}$ (see Kalnins (1986) [19]). There has been considerable
work done in two and three dimensions, however there still remains a lot of
work to be done for a detailed analysis of fundamental solutions.

We define an unnormalized fundamental solution of Laplace’s equation on the
unit hyperboloid
${\mathfrak{h}}^{d}:({\mathbf{H}}^{d}\times{\mathbf{H}}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbf{H}}^{d}\\}\to{\mathbf{R}}$ such that

${\mathfrak{h}}^{d}({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime}):={\mathcal{I}}_{d}(\rho({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime}))=\frac{2\pi^{d/2}R^{d-2}}{\Gamma(d/2)}{\mathcal{H}}_{R}^{d}({\bf
x},{{\bf x}^{\prime}}).$ (32)

In our current azimuthal Fourier analysis, we therefore will focus on the
relatively easier case of separable subgroup-type coordinate systems on
${\mathbf{H}}_{R}^{d}$, and specifically for geodesic polar coordinates. In
these coordinates the Riemannian metric is given by (9) and we further
restrict our attention by adopting standard geodesic polar coordinates (1).

In these coordinates would would like to expand a fundamental solution of
Laplace’s equation on the hyperboloid in an azimuthal Fourier series, namely

${\mathfrak{h}}^{d}({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime})=\sum_{m=0}^{\infty}\cos(m(\phi-\phi^{\prime})){\sf
H}_{m}^{d/2-1}(r,r^{\prime},\theta_{1},\ldots,\theta_{d-2},\theta_{1}^{\prime},\ldots,\theta_{d-2}^{\prime})$
(33)

where ${\sf H}_{m}^{d/2-1}:[0,\infty)^{2}\times[0,\pi]^{2d-4}\to{\mathbf{R}}$
is defined such that

${\sf
H}_{m}^{d/2-1}(r,r^{\prime},\theta_{1},\ldots,\theta_{d-2},\theta_{1}^{\prime},\ldots,\theta_{d-2}^{\prime}):=\frac{\epsilon_{m}}{\pi}\int_{0}^{\pi}{\mathfrak{h}}^{d}({\widehat{\bf
x}},{\widehat{\bf
x}}^{\prime})\cos(m(\phi-\phi^{\prime}))d(\phi-\phi^{\prime})$ (34)

(see for instance Cohl & Tohline (1999) [10]). According to Theorem 2.1 and
(32), we may write ${\mathfrak{h}}^{d}({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime})$ in terms of associated Legendre functions of the second kind as
follows

${\mathfrak{h}}^{d}({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime})=\frac{e^{-i\pi(d/2-1)}}{2^{d/2-1}\Gamma(d/2)\,(\sinh
d({\widehat{\bf x}},{\widehat{\bf
x}}^{\prime}))^{d/2-1}}Q_{d/2-1}^{d/2-1}\left(\cosh d({\widehat{\bf
x}},{\widehat{\bf x}}^{\prime})\right).$ (35)

By (3) we know that in any geodesic polar coordinate system

$\cosh d({\widehat{\bf x}},{\widehat{\bf x}}^{\prime})=\cosh r\cosh
r^{\prime}-\sinh r\sinh r^{\prime}\cos\gamma,$ (36)

and therefore through (34), (35), and (36), in standard geodesic polar
coordinates, the azimuthal Fourier coefficient can be given by

$\eqalign{{\sf
H}_{m}^{d/2-1}(r,r^{\prime},\theta_{1},\ldots,\theta_{d-2},\theta_{1}^{\prime},\ldots,\theta_{d-2}^{\prime})\cr\hskip
71.13188pt=\frac{\epsilon_{m}e^{-i\pi(d/2-1)}}{2^{d/2-1}\pi\Gamma(d/2)}\int_{0}^{\pi}\frac{Q_{d/2-1}^{d/2-1}\left(A-B\cos\psi\right)\cos(m\psi)}{\left[(A-B\cos\psi)^{2}-1\right]^{(d-2)/4}}d\psi,}$
(37)

where $\psi:=\phi-\phi^{\prime},$
$A,B:[0,\infty)^{2}\times[0,\pi]^{2d-4}\to{\mathbf{R}}$ are defined through
(8) and (36) as

$\displaystyle
A(r,r^{\prime},\theta_{1},\ldots,\theta_{d-2},\theta_{1}^{\prime},\ldots,\theta_{d-2}^{\prime}):=\cosh
r\cosh
r^{\prime}\sum_{i=1}^{d-2}\cos\theta_{i}{\cos\theta_{i}}^{\prime}\prod_{j=1}^{i-1}\sin\theta_{j}{\sin\theta_{j}}^{\prime},$

and

$\displaystyle
B(r,r^{\prime},\theta_{1},\ldots,\theta_{d-2},\theta_{1}^{\prime},\ldots,\theta_{d-2}^{\prime}):=\sinh
r\sinh r^{\prime}\prod_{i=1}^{d-2}\sin\theta_{i}{\sin\theta_{i}}^{\prime}.$

Even though (37) is a compact expression for the Fourier coefficient of a
fundamental solution of Laplace’s equation on ${\mathbf{H}}_{R}^{d}$ for
$d\in\\{2,3,4,\ldots\\},$ it may be informative to use any of the
representations of a fundamental solution of the Laplacian on
${\mathbf{H}}_{R}^{d}$ from Theorem 2.1 to express the Fourier coefficients.
For instance if one uses the finite-summation expression in the odd-
dimensions, on can write the Fourier coefficients as a linear combination of
integrals of the form

$\int_{-1}^{1}\frac{\left[(a+b)/2-x\right]^{2k-1}x^{p}dx}{(a-x)^{k-1}(b-x)^{k-1}\sqrt{(a-x)(b-x)(c-x)(x-d)}},$

where $x=\cos\psi$, $k\in\\{1,\ldots,(d-1)/2\\}$, $p\in\\{0,\ldots,m\\},$ and
we have used the nomenclature of section 3.2. This integral is a rational
function of $x$ multiplied by an inverse square-root of a quartic in $x$.
Because of this and due to the limits of integration, we see that by
definition, these are all given in terms of complete elliptic integrals. The
special functions which represent the azimuthal Fourier coefficients on
${\mathbf{H}}_{R}^{d}$ are unlike the odd-half-integer degree, integer-order,
associated Legendre functions of the second kind which appear in Euclidean
space ${\mathbf{R}}^{d}$ for $d$ odd (see Cohl (2010) [6]; Cohl & Dominici
(2010) [7]), in that they include complete elliptic integrals of the third
kind (in addition to complete elliptic integrals of the first and second kind)
(cf. section 3.2) in their basis functions. For $d\geq 2$, through (4.1) in
Cohl & Dominici (2010) [7] and that ${\mathbf{H}}_{R}^{d}$ is a manifold (and
therefore must locally represent Euclidean space), the functions ${\sf
H}_{m}^{d/2-1}$ are generalizations of associated Legendre functions of the
second kind with odd-half-integer degree and order given by either an odd-
half-integer or an integer.

##  4 Gegenbauer expansion in geodesic polar coordinates

In this section we derive an eigenfunction expansion for a fundamental
solution of Laplace’s equation on the hyperboloid in geodesic polar
coordinates for $d\in\\{3,4,\ldots\\}.$ Since the spherical harmonics for
$d=2$ are just trigonometric functions with argument given in terms of the
azimuthal angle, this case has already been covered in section 3.1.

In geodesic polar coordinates, Laplace’s equation is given by (cf. (10))

$\Delta f=\frac{1}{R^{2}}\left[\frac{\partial^{2}f}{\partial r^{2}}+(d-1)\coth
r\frac{\partial f}{\partial
r}+\frac{1}{\sinh^{2}r}\Delta_{{\mathbf{S}}^{d-1}}\right]f=0,$ (38)

where $f:{\mathbf{H}}_{R}^{d}\to{\mathbf{R}}$ and
$\Delta_{{\mathbf{S}}^{d-1}}$ is the corresponding Laplace-Beltrami operator
on the $(d-1)$-dimensional unit sphere ${\mathbf{S}}^{d-1}$. Eigenfunctions
$Y_{l}^{K}:{\mathbf{S}}^{d-1}\to{\mathbf{C}}$ of the Laplace-Beltrami operator
$\Delta_{{\mathbf{S}}^{d-1}}$, where $l\in{\mathbf{N}}_{0}$ and $K$ is a set
of quantum numbers which label representations for $l$ in separable subgroup
type coordinate systems on ${\mathbf{S}}^{d-1}$ (i.e. angular momentum type
quantum numbers, see Izmest’ev et al. (2001) [18]), are given by solutions to
the eigenvalue problem (11).

In standard geodesic polar coordinates (1),
$K=(k_{1},\ldots,k_{d-3},|k_{d-2}|)\in{\mathbf{N}}_{0}^{d-2}$ with
$k_{0}=l\geq k_{1}\geq\ldots\geq k_{d-3}\geq|k_{d-2}|\geq 0$, and in
particular $k_{d-2}\in\\{-k_{d-3},\ldots,k_{d-3}\\}$. A positive fundamental
solution
${\mathcal{H}}_{R}^{d}:({\mathbf{H}}_{R}^{d}\times{\mathbf{H}}_{R}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbf{H}}_{R}^{d}\\}\to{\mathbf{R}}$ on the
$R$-radius hyperboloid satisfies (12). The completeness relation for
hyperspherical harmonics in standard hyperspherical coordinates is given by

$\sum_{l=0}^{\infty}\sum_{K}Y_{l}^{K}(\theta_{1},\ldots,\theta_{d-1})\overline{Y_{l}^{K}(\theta_{1}^{\prime},\ldots,\theta_{d-1}^{\prime})}=\frac{\delta(\theta_{1}-\theta_{1}^{\prime})\ldots\delta(\theta_{d-1}-\theta_{d-1}^{\prime})}{\sin^{d-2}\theta_{d-1}^{\prime}\ldots\sin\theta_{2}^{\prime}},$

where $K=(k_{1},\ldots,k_{d-2})$ and $l=k_{0}\in{\mathbf{N}}_{0}$. Therefore
through (14), we can write

$\delta_{g}({\bf x},{{\bf
x}^{\prime}})=\frac{\delta(r-r^{\prime})}{R^{d}\sinh^{d-1}r^{\prime}}\sum_{l=0}^{\infty}\sum_{K}Y_{l}^{K}(\theta_{1},\ldots,\theta_{d-1})\overline{Y_{l}^{K}(\theta_{1}^{\prime},\ldots,\theta_{d-1}^{\prime})}.$
(39)

For fixed $r,r^{\prime}\in[0,\infty)$ and
$\theta_{1}^{\prime},\ldots,\theta_{d-1}^{\prime}\in[0,\pi]$, since
${\mathcal{H}}_{R}^{d}$ is harmonic on its domain, its restriction is in
$C^{2}({\mathbf{S}}^{d-1})$, and therefore has a unique expansion in
hyperspherical harmonics, namely

${\mathcal{H}}_{R}^{d}({\bf x},{{\bf
x}^{\prime}})=\sum_{l=0}^{\infty}\sum_{K}u_{l}^{K}(r,r^{\prime},\theta_{1}^{\prime},\ldots,\theta_{d-1}^{\prime})Y_{l}^{K}(\theta_{1},\ldots,\theta_{d-1}),$
(40)

where $u_{l}^{K}:[0,\infty)^{2}\times[0,\pi]^{d-1}\to{\mathbf{C}}$. If we
substitute (39) and (40) into (12) and use (38) and (11), we obtain

$\displaystyle\sum_{l=0}^{\infty}\sum_{K}Y_{l}^{K}(\theta_{1},\ldots,\theta_{d-1})\left[\frac{d^{2}}{dr^{2}}+(d-1)\coth
r\frac{d}{dr}-\frac{l(l+d-2)}{\sinh^{2}r}\right]u_{l}^{K}(r,r^{\prime},\theta_{1}^{\prime},\dots,\theta_{d-1}^{\prime})$
$\displaystyle\hskip
5.69046pt=\sum_{l=0}^{\infty}\sum_{K}Y_{l}^{K}(\theta_{1},\ldots,\theta_{d-1})\overline{Y_{l}^{K}(\theta_{1}^{\prime},\ldots,\theta_{d-1}^{\prime})}\cdot\frac{\delta(r-r^{\prime})}{R^{d-2}\sinh^{d-1}r^{\prime}}.$
(41)

This indicates that for $u_{l}:[0,\infty)^{2}\to{\mathbf{R}}$,

$u_{l}^{K}(r,r^{\prime},\theta_{1}^{\prime},\ldots,\theta_{d-1}^{\prime})=u_{l}(r,r^{\prime})\overline{Y_{l}^{K}(\theta_{1}^{\prime},\ldots,\theta_{d-1}^{\prime})},$
(42)

and from (40) the expression for a fundamental of the Laplace-Beltrami
operator in hyperspherical coordinates on the hyperboloid is given by

${\mathcal{H}}_{R}^{d}({\bf x},{{\bf
x}^{\prime}})=\sum_{l=0}^{\infty}u_{l}(r,r^{\prime})\sum_{K}Y_{l}^{K}(\theta_{1},\ldots,\theta_{d-1})\overline{Y_{l}^{K}(\theta_{1}^{\prime},\ldots,\theta_{d-1}^{\prime})}.$
(43)

The above expression can be simplified using the addition theorem for
hyperspherical harmonics (see for instance Wen & Avery (1985) [30], section
10.2.1 in Fano & Rau (1996), Chapter 9 in Andrews, Askey & Roy (1999) [2] and
especially Chapter XI in Erdélyi et al. Vol. II (1981) [12]), which is given
by

$\sum_{K}Y_{l}^{K}({\widehat{\bf x}})\overline{Y_{l}^{K}({\widehat{\bf
x}}^{\prime})}=\frac{\Gamma(d/2)}{2\pi^{d/2}(d-2)}(2l+d-2)C_{l}^{d/2-1}(\cos\gamma),$
(44)

where $\gamma$ is the angle between two arbitrary vectors ${\widehat{\bf
x}},{\widehat{\bf x}}^{\prime}\in{\mathbf{S}}^{d-1}$ given in terms of (2).
The Gegenbauer polynomials $C_{l}^{\mu}:[-1,1]\to{\mathbf{R}}$,
$l\in{\mathbf{N}}_{0}$, $\mbox{Re}\,\mu>-1/2$, can be defined in terms of the
Gauss hypergeometric function as

$C_{l}^{\mu}(x):=\frac{(2\mu)_{l}}{l!}\,{}_{2}F_{1}\left(-l,l+2\mu;\mu+\frac{1}{2};\frac{1-x}{2}\right).$

The above expression (43) can be simplified using (44), therefore

${\mathcal{H}}_{R}^{d}({\bf x},{{\bf
x}^{\prime}})=\frac{\Gamma(d/2)}{2\pi^{d/2}(d-2)}\sum_{l=0}^{\infty}u_{l}(r,r^{\prime})(2l+d-2)C_{l}^{d/2-1}(\cos\gamma).$
(45)

Now we compute the exact expression for $u_{l}(r,r^{\prime})$. By separating
the angular dependence in (41) and using (42), we obtain the differential
equation

$\frac{d^{2}u_{l}}{dr^{2}}+(d-1)\coth
r\frac{du_{l}}{dr}-\frac{l(l+d-2)u_{l}}{\sinh^{2}r}=-\frac{\delta(r-r^{\prime})}{R^{d-2}\sinh^{d-1}r^{\prime}}.$
(46)

Away from $r=r^{\prime}$, solutions to the differential equation (46) must be
given by solutions to the homogeneous equation, which are are given in section
2.2. Therefore, the solution to (46) is given by

$u_{l}(r,r^{\prime})=\frac{A}{\left(\sinh r\sinh
r^{\prime}\right)^{d/2-1}}P_{d/2-1}^{-(d/2-1+l)}(\cosh
r_{<})Q_{d/2-1}^{d/2-1+l}(\cosh r_{>}),$ (47)

such that $u_{l}(r,r^{\prime})$ is continuous at $r=r^{\prime}$, where
$A\in{\mathbf{R}}.$

In order to determine the constant $A$, we first make the substitution

$v_{l}(r,r^{\prime})=(\sinh r\sinh r^{\prime})^{(d-1)/2}u_{l}(r,r^{\prime}).$
(48)

This converts (46) into the following differential equation

$\frac{\partial^{2}v_{l}(r,r^{\prime})}{\partial
r^{2}}-\frac{1}{4}\left[\frac{(d-1+2l)(d-3+2l)}{\sinh^{2}r}+(d-1)^{2}\right]v_{l}(r,r^{\prime})=-\frac{\delta(r-r^{\prime})}{R^{d-2}},$

which we then integrate over $r$ from $r^{\prime}-\epsilon$ to
$r^{\prime}+\epsilon$, and take the limit as $\epsilon\to 0^{+}$. We are left
with a discontinuity condition for the derivative of $v_{l}(r,r^{\prime})$
with respect to $r$ evaluated at $r=r^{\prime}$, namely

$\lim_{r\to
0^{+}}\left.\frac{dv_{l}(r,r^{\prime})}{dr}\right|_{r^{\prime}-\epsilon}^{r^{\prime}+\epsilon}=\frac{-1}{R^{d-2}}.$
(49)

After inserting (47) with (48) into (49), substituting $z=\cosh r^{\prime}$,
evaluating at $r=r^{\prime}$, and making use of the Wronskian relation (e.g.
p. 165 in Magnus, Oberhettinger & Soni (1966) [21])

$W\left\\{P_{\nu}^{-\mu}(z),Q_{\nu}^{\mu}(z)\right\\}=-\frac{e^{i\pi\mu}}{z^{2}-1},$

which is equivalent to

$W\left\\{P_{\nu}^{-\mu}(\cosh r^{\prime}),Q_{\nu}^{\mu}(\cosh
r^{\prime})\right\\}=-\frac{e^{i\pi\mu}}{\sinh^{2}r^{\prime}},$

we obtain

$A=\frac{e^{-i\pi(d/2-1+l)}}{R^{d-2}},$

and hence

$u_{l}(r,r^{\prime})=\frac{e^{-i\pi(d/2-1+l)}}{R^{d-2}(\sinh r\sinh
r^{\prime})^{d/2-1}}P_{d/2-1}^{-(d/2-1+l)}(\cosh
r_{<})Q_{d/2-1}^{d/2-1+l}(\cosh r_{>}),$

and therefore through (45), we have

$\displaystyle{\mathcal{H}}_{R}^{d}({\bf x},{{\bf
x}^{\prime}})=\frac{\Gamma(d/2)}{2\pi^{d/2}R^{d-2}(d-2)}\frac{e^{-i\pi(d/2-1)}}{(\sinh
r\sinh r^{\prime})^{d/2-1}}$
$\displaystyle\times\sum_{l=0}^{\infty}(-1)^{l}(2l+d-2)P_{d/2-1}^{-(d/2-1+l)}(\cosh
r_{<})Q_{d/2-1}^{d/2-1+l}(\cosh r_{>})C_{l}^{d/2-1}(\cos\gamma).$ (50)

As an alternative check of our derivation, we can do the asymptotics for the
product of associated Legendre functions $P_{d/2-1}^{-(d/2-1+l)}(\cosh
r_{<})Q_{d/2-1}^{d/2-1+l}(\cosh r_{>})$ in (50) as $r,r^{\prime}\to 0^{+}$.
The appropriate asymptotic expressions for $P$ and $Q$ respectively can be
found on p. 171 and p. 173 in Olver (1997) [24]. For the associated Legendre
function of the first kind there is

$P_{\nu}^{-\mu}(z)\sim\frac{\left[(z-1)/2\right]^{\mu/2}}{\Gamma(\mu+1)},$

as $z\to 1$, $\mu\neq-1,-2,\ldots$, and for the associated Legendre function
of the second kind there is

$Q_{\nu}^{\mu}(z)\sim\frac{e^{i\pi\mu}\Gamma(\mu)}{2\left[(z-1)/2\right]^{\mu/2}},$

as $z\to 1^{+}$, $\mbox{Re}\ \,\mu>0$, and $\nu+\mu\neq-1,-2,-3,\ldots$. To
second order the hyperbolic cosine is given by $\cosh r\simeq 1+r^{2}/2$.
Therefore to lowest order we can insert $\cosh r_{<}\simeq 1+r_{<}^{2}/2$ and
$\cosh r_{>}\simeq 1+r_{>}^{2}/2$ into the above expressions yielding

$P_{d/2-1}^{-(d/2-1+l)}(\cosh
r_{<})\sim\frac{(r_{<}/2)^{d/2-1+l}}{\Gamma(d/2+l)},$

and

$Q_{d/2-1}^{d/2-1+l}(\cosh
r_{>})\sim\frac{e^{i\pi(d/2-1+l)}\Gamma(d/2-1+l)}{2(r_{>}/2)^{d/2-1+l}},$

as $r,r^{\prime}\to 0^{+}$. Therefore the asymptotics for the product of
associated Legendre functions in (50) is given by

$P_{d/2-1}^{-(d/2-1+l)}(\cosh r_{<})Q_{d/2-1}^{d/2-1+l}(\cosh
r_{>})\sim\frac{e^{i\pi(d/2-1+l)}}{2l+d-2}\left(\frac{r_{<}}{r_{>}}\right)^{l+d/2-1}$
(51)

(the factor $2l+d-2$ is a term which one encounters regularly with
hyperspherical harmonics). Gegenbauer polynomials obey the following
generating function

$\frac{1}{\left(1+z^{2}-2zx\right)^{\mu}}=\sum_{l=0}^{\infty}C_{l}^{\mu}(x)z^{l},$
(52)

where $x\in[-1,1]$ and $|z|<1$ (see for instance, p. 222 in Magnus,
Oberhettinger & Soni (1966) [21]). The generating function for Gegenbauer
polynomials (52) can be used to expand a fundamental solution of Laplace’s
equation in Euclidean space ${\mathbf{R}}^{d}$ (for $d\geq 3$, cf. Theorem
3.1) in hyperspherical coordinates, namely

$\frac{1}{\|{\bf x}-{{\bf
x}^{\prime}}\|^{d-2}}=\sum_{l=0}^{\infty}\frac{r_{<}^{l}}{r_{>}^{l+d-2}}C_{l}^{d/2-1}(\cos\gamma),$
(53)

where $\gamma$ was defined in (44). Using (53) and Theorem 3.1, since
${\mathcal{H}}_{R}^{d}\to{\mathcal{G}}^{d}$, $\sinh r,\sinh r^{\prime}\to
r,r^{\prime}$ and (51) is satisfied to lowest order as $r,r^{\prime}\to
0^{+}$, we see that (50) obeys the correct asymptotics and our fundamental
solution expansion locally reduces to the appropriate expansion for Euclidean
space, as it should since ${\mathbf{H}}_{R}^{d}$ is a manifold.

Note that (50) can be further expanded over the remaining $(d-2)$-quantum
numbers in $K$ in terms of a simply separable product of normalized harmonics
$Y_{l}^{K}({\widehat{\bf x}})\overline{Y_{l}^{K}({\widehat{\bf
x}}^{\prime})}$, where ${\widehat{\bf x}},{\widehat{\bf
x}}^{\prime}\in{\mathbf{S}}^{d-1}$, using the addition theorem for
hyperspherical harmonics (44) (see Cohl (2010) [6] for several examples).

It is intriguing to observe how one might obtain the Fourier expansion for
$d=2$ (24) from the the expansion (50), which is strictly valid for $d\geq 3$.
If one makes the substitution $\mu=d/2-1$ in (50) then we obtain the following
conjecture (which matches up to the generating function for Gegenbauer
polynomials in the Euclidean limit $r,r^{\prime}\to 0^{+}$)

$\displaystyle\frac{1}{\sinh^{\mu}\rho}Q_{\mu}^{\mu}(\cosh\rho)$
$\displaystyle=$ $\displaystyle\frac{2^{\mu}\Gamma(\mu+1)}{(\sinh r\sinh
r^{\prime})^{\mu}}$ (54) $\displaystyle\hskip
5.69046pt\times\sum_{n=0}^{\infty}(-1)^{n}\frac{n+\mu}{\mu}P_{\mu}^{-(\mu+n)}(\cosh
r_{<})Q_{\mu}^{\mu+n}(\cosh r_{>})C_{n}^{\mu}(\cos\gamma),$

for all $\mu\in{\mathbf{C}}$ such that $\mbox{Re}\,\mu>-1/2$. If we take the
limit as $\mu\to 0$ in (54) and use

$\lim_{\mu\to 0}\frac{{n}+\mu}{\mu}C_{n}^{\mu}(x)=\epsilon_{n}T_{n}(x)$ (55)

(see for instance (6.4.13) in Andrews, Askey & Roy (1999) [2]), where
$T_{n}:[-1,1]\to{\mathbf{R}}$ is the Chebyshev polynomial of the first kind
defined as $T_{l}(x):=\cos(l\cos^{-1}x),$ then we obtain the following formula

$\frac{1}{2}\log\frac{\cosh\rho+1}{\cosh\rho-1}=\sum_{n=0}^{\infty}\epsilon_{n}(-1)^{n}P_{0}^{-n}(\cosh
r_{<})Q_{0}^{n}(\cosh r_{>})\cos(n(\phi-\phi^{\prime})),$

where $\cosh\rho=\cosh r\cosh r^{\prime}-\sinh r\sinh
r^{\prime}\cos(\phi-\phi^{\prime})$. By taking advantage of the following
formulae

$P_{0}^{-n}(z)=\frac{1}{n!}\left[\frac{z-1}{z+1}\right]^{n/2},$ (56)

for $n\geq 0,$

$Q_{0}(z)=\frac{1}{2}\log\frac{z+1}{z-1}$ (57)

(see (8.4.2) in Abramowitz & Stegun (1972) [1]), and

$Q_{0}^{n}(z)=\frac{1}{2}(-1)^{n}(n-1)!\left\\{\left[\frac{z+1}{z-1}\right]^{n/2}-\left[\frac{z-1}{z+1}\right]^{n/2}\right\\},$
(58)

for $n\geq 1$ then (24) is reproduced. The representation (56) follows easily
from the Gauss hypergeometric representation of the associated Legendre
function of the first kind (see (8.1.2) in Abramowitz & Stegun (1972) [1])

$P_{\nu}^{\mu}(z)=\frac{1}{\Gamma(1-\mu)}\left[\frac{z+1}{z-1}\right]^{\mu/2}{}_{2}F_{1}\left(-\nu,\nu+1;1-\mu;\frac{1-z}{2}\right).$
(59)

One way to derive the representation of the associated Legendre function of
second kind (58) is to use Whipple formula for associated Legendre functions
(cf. (8.2.7) in Abramowitz & Stegun (1972) [1])

$Q_{\nu}^{\mu}(z)=\sqrt{\pi}{2}\Gamma(\nu+\mu+1)(z^{2}-1)^{-1/4}e^{i\pi\mu}P_{-\mu-1/2}^{-\nu-1/2}\left(\frac{z}{\sqrt{z^{2}-1}}\right),$

and (8.6.9) in Abramowitz & Stegun (1972) [1], namely

$P_{\nu}^{-1/2}(z)=\sqrt{\frac{2}{\pi}}\frac{(z^{2}-1)^{-1/4}}{2\nu+1}\left\\{\left[z+\sqrt{z^{2}-1}\right]^{\nu+1/2}-\left[z+\sqrt{z^{2}-1}\right]^{-\nu-1/2}\right\\},$

for $\nu\neq-1/2$.

###  4.1 Addition theorem for the azimuthal Fourier coefficient on
${\mathbf{H}}_{R}^{3}$

One can compute addition theorems for the azimuthal Fourier coefficients of a
fundamental solution for Laplace’s equation on ${\mathbf{H}}_{R}^{d}$ for
$d\geq 3$ by relating directly obtained Fourier coefficients to the expansion
over hyperspherical harmonics for the same fundamental solution. By using the
expansion of ${\mathcal{H}}_{R}^{d}({\bf x},{{\bf x}^{\prime}})$ in terms of
Gegenbauer polynomials (50) in combination with the addition theorem for
hyperspherical harmonics (44) expressed in, for instance, one of Vilenkin’s
polyspherical coordinates (see section IX.5.2 in Vilenkin (1968) [29];
Izmest’ev et al. (1999,2001) [17, 18]), one can obtain through series
rearrangement a multi-summation expression for the azimuthal Fourier
coefficients. Vilenkin’s polyspherical coordinates are simply subgroup-type
coordinate systems which parametrize points on ${\mathbf{S}}^{d-1}$ (for a
detailed discussion of these coordinate systems see chapter 4 in Cohl (2010)
[6]). In this section we will give an explicit example of just such an
addition theorem on ${\mathbf{H}}_{R}^{3}$.

The azimuthal Fourier coefficients on ${\mathbf{H}}_{R}^{3}$ expressed in
standard hyperspherical coordinates (1) are given by the functions ${\sf
H}_{m}:[0,\infty)^{2}\times[0,\pi]^{2}\to{\mathbf{R}}$ which is defined by
(28). By expressing (50) in $d=3$ we obtain

$\displaystyle{\mathcal{H}}_{R}^{3}({\bf x},{{\bf x}^{\prime}})=\frac{-i}{4\pi
R\sqrt{\sinh r\sinh r^{\prime}}}$ $\displaystyle\hskip
11.38092pt\times\sum_{l=0}^{\infty}(-1)^{l}(2l+1)P_{1/2}^{-(1/2+l)}(\cosh
r_{<})Q_{1/2}^{1/2+l}(\cosh r_{>})P_{l}(\cos\gamma),$ (60)

where $P_{l}:[-1,1]\to{\mathbf{R}}$ is the Legendre polynomial defined by
$P_{l}(x)=C_{l}^{1/2}(x)$, or through (59) with $\mu=0$ and
$\nu\in{\mathbf{N}}_{0}$. By using the addition theorem for hyperspherical
harmonics (44) with $d=3$ using
$(\cos\theta,\sin\theta\cos\phi,\sin\theta\sin\phi)$ to parametrize points on
${\mathbf{S}}^{2},$ we have since the normalized spherical harmonics are

$Y_{l,m}(\theta,\phi)=(-1)^{m}\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}P_{l}^{m}(\cos\theta)e^{im\phi},$

the addition theorem for spherical harmonics, namely

$P_{l}(\cos\gamma)=\sum_{m=-l}^{l}\frac{(l-m)!}{(l+m)!}P_{l}^{m}(\cos\theta)P_{l}^{m}(\cos\theta^{\prime})e^{im(\phi-\phi^{\prime})},$
(61)

where $\cos\gamma$ is given by (26). By combining (60) and (61), reversing the
order of the two summation symbols, and comparing the result with (27) we
obtain the following single summation addition theorem for the azimuthal
Fourier coefficients of a fundamental solution of Laplace’s equation on
${\mathbf{H}}_{R}^{3}$, namely since ${\mathfrak{h}}^{3}=4\pi
R{\mathcal{H}}_{R}^{3}$,

$\displaystyle{\sf
H}_{m}^{1/2}(r,r^{\prime},\theta,\theta^{\prime})=\frac{-i\epsilon_{m}}{\sqrt{\sinh
r\sinh r^{\prime}}}\sum_{l=|m|}^{\infty}(-1)^{l}(2l+1)\frac{(l-m)!}{(l+m)!}$
$\displaystyle\hskip 56.9055pt\times
P_{l}^{m}(\cos\theta)P_{l}^{m}(\cos\theta^{\prime})P_{1/2}^{-(1/2+l)}(\cosh
r_{<})Q_{1/2}^{1/2+l}(\cosh r_{>}).$

This addition theorem reduces to the corresponding result ((2.4) in Cohl et
al. (2001) [9]) in the Euclidean ${\mathbf{R}}^{3}$ limit as $r,r^{\prime}\to
0^{+}$.

##  5 Discussion

Re-arrangement of the multi-summation expressions in section 4 is possible
through modification of the order in which the countably infinite space of
quantum numbers is summed over in a standard hyperspherical coordinate system,
namely

$\displaystyle\sum_{l=0}^{\infty}\sum_{K}=\sum_{l=0}^{\infty}\sum_{k_{1}=0}^{l}\sum_{k_{2}=0}^{k_{1}}\cdots\sum_{k_{d-4}=0}^{k_{d-5}}\sum_{k_{d-3}=0}^{k_{d-4}}\sum_{k_{d-2}=-k_{d-3}}^{k_{d-3}}$
$\displaystyle\hskip
39.83368pt=\sum_{k_{d-2}=-\infty}^{\infty}\sum_{k_{d-3}=|k_{d-2}|}^{\infty}\sum_{k_{d-4}=k_{d-2}}^{\infty}\cdots\sum_{k_{2}=k_{3}}^{\infty}\sum_{k_{1}=k_{2}}^{\infty}\sum_{k_{0}=k_{1}}^{\infty}.$

Similar multi-summation re-arrangements have been accomplished previously for
azimuthal Fourier coefficients of fundamental solutions for the Laplacian in
Euclidean space (see for instance Cohl et al. (2000) [11]; Cohl et al. (2001)
[9]). Comparison of the azimuthal Fourier expansions in section 3 (and in
particular (37)) with re-arranged Gegenbauer expansions in section 4 (and in
particular (50)) will yield new addition theorems for the special functions
representing the azimuthal Fourier coefficients of a fundamental solution of
the Laplacian on the hyperboloid. These implied addition theorems will provide
new special function identities for the azimuthal Fourier coefficients, which
are hyperbolic generalizations of particular associated Legendre functions of
the second kind. In odd-dimensions, these special functions reduce to toroidal
harmonics.

## Acknowledgements

Much thanks to A. Rod Gover, Tom ter Elst, Shaun Cooper, and Willard Miller,
Jr. for valuable discussions. I would like to express my gratitude to Carlos
Criado Cambón in the Facultad de Ciencias at Universidad de Málaga for his
assistance in describing the global geodesic distance function in the
hyperboloid model. We would also like to acknowledge two anonymous referees
whose comments helped improve this paper. I acknowledge funding for time to
write this paper from the Dean of the Faculty of Science at the University of
Auckland in the form of a three month stipend to enhance University of
Auckland 2012 PBRF Performance. Part of this work was conducted while H. S.
Cohl was a National Research Council Research Postdoctoral Associate in the
Information Technology Laboratory at the National Institute of Standards and
Technology, Gaithersburg, Maryland, U.S.A.

## References

## References

  * [1] M. Abramowitz and I. A. Stegun.  Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55 of National Bureau of Standards Applied Mathematics Series.  U.S. Government Printing Office, Washington, D.C., 1972.
  * [2] G. E. Andrews, R. Askey, and R. Roy.  Special functions, volume 71 of Encyclopedia of Mathematics and its Applications.  Cambridge University Press, Cambridge, 1999.
  * [3] E. Beltrami.  Essai d’interprétation de la géométrie noneuclidéenne. Trad. par J. Hoüel.  Annales Scientifiques de l’École Normale Supérieure, 6:251–288, 1869.
  * [4] L. Bers, F. John, and M. Schechter.  Partial differential equations.  Interscience Publishers, New York, N.Y., 1964.
  * [5] P. F. Byrd and M. D. Friedman.  Handbook of elliptic integrals for engineers and physicists.  Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Bd LXVII. Springer-Verlag, Berlin, 1954.
  * [6] H. S. Cohl.  Fourier and Gegenbauer expansions for fundamental solutions of the Laplacian and powers in ${\mathbf{R}}^{d}$ and ${\mathbf{H}}^{d}$.  PhD thesis, The University of Auckland, 2010.  xiv+190 pages.
  * [7] H. S. Cohl and D. E. Dominici.  Generalized Heine’s identity for complex Fourier series of binomials.  Proceedings of the Royal Society A, 467:333–345, 2010.
  * [8] H. S. Cohl and E. G. Kalnins.  Fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry.  In review, Journal of Physics A: Mathematical and Theoretical, 2011.
  * [9] H. S. Cohl, A. R. P. Rau, J. E. Tohline, D. A. Browne, J. E. Cazes, and E. I. Barnes.  Useful alternative to the multipole expansion of $1/r$ potentials.  Physical Review A: Atomic and Molecular Physics and Dynamics, 64(5):052509, Oct 2001.
  * [10] H. S. Cohl and J. E. Tohline.  A Compact Cylindrical Green’s Function Expansion for the Solution of Potential Problems.  The Astrophysical Journal, 527:86–101, December 1999.
  * [11] H. S. Cohl, J. E. Tohline, A. R. P. Rau, and H. M. Srivastava.  Developments in determining the gravitational potential using toroidal functions.  Astronomische Nachrichten, 321(5/6):363–372, 2000.
  * [12] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi.  Higher transcendental functions. Vol. II.  Robert E. Krieger Publishing Co. Inc., Melbourne, Fla., 1981.
  * [13] G. B. Folland.  Introduction to partial differential equations.  Number 17 in Mathematical Notes. Princeton University Press, Princeton, 1976.
  * [14] L. Fox and I. B. Parker.  Chebyshev polynomials in numerical analysis.  Oxford University Press, London, 1968.
  * [15] D. Gilbarg and N. S. Trudinger.  Elliptic partial differential equations of second order.  Number 224 in Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin etc., second edition, 1983.
  * [16] C. Grosche, G. S. Pogosyan, and A. N. Sissakian.  Path-integral approach for superintegrable potentials on the three-dimensional hyperboloid.  Physics of Particles and Nuclei, 28(5):486–519, 1997.
  * [17] A. A. Izmest’ev, G. S. Pogosyan, A. N. Sissakian, and P. Winternitz.  Contractions of Lie algebras and separation of variables. The $n$-dimensional sphere.  Journal of Mathematical Physics, 40(3):1549–1573, 1999.
  * [18] A. A. Izmest’ev, G. S. Pogosyan, A. N. Sissakian, and P. Winternitz.  Contractions of Lie algebras and the separation of variables: interbase expansions.  Journal of Physics A: Mathematical and General, 34(3):521–554, 2001\.
  * [19] E. G. Kalnins.  Separation of variables for Riemannian spaces of constant curvature, volume 28 of Pitman Monographs and Surveys in Pure and Applied Mathematics.  Longman Scientific & Technical, Harlow, 1986.
  * [20] J. M. Lee.  Riemannian manifolds, volume 176 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1997.
  * [21] W. Magnus, F. Oberhettinger, and R. P. Soni.  Formulas and theorems for the special functions of mathematical physics.  Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52. Springer-Verlag New York, Inc., New York, 1966.
  * [22] P. M. Morse and H. Feshbach.  Methods of theoretical physics. 2 volumes.  McGraw-Hill Book Co., Inc., New York, 1953.
  * [23] M. N. Olevskiĭ.  Triorthogonal systems in spaces of constant curvature in which the equation $\Delta_{2}u+\lambda u=0$ allows a complete separation of variables.  Matematicheskiĭ Sbornik, 27(69):379–426, 1950.  (in Russian).
  * [24] F. W. J. Olver.  Asymptotics and special functions.  AKP Classics. A K Peters Ltd., Wellesley, MA, 1997.  Reprint of the 1974 original [Academic Press, New York].
  * [25] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors.  NIST handbook of mathematical functions.  Cambridge University Press, Cambridge, 2010.
  * [26] G. S. Pogosyan and P. Winternitz.  Separation of variables and subgroup bases on $n$-dimensional hyperboloids.  Journal of Mathematical Physics, 43(6):3387–3410, 2002.
  * [27] W. P. Thurston.  Three-dimensional geometry and topology. Vol. 1, volume 35 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 1997.  Edited by Silvio Levy.
  * [28] R. J. Trudeau.  The non-Euclidean revolution.  Birkhäuser, Boston, 1987.
  * [29] N. Ja. Vilenkin.  Special functions and the theory of group representations.  Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, Vol. 22. American Mathematical Society, Providence, R. I., 1968.
  * [30] Z. Y. Wen and J. Avery.  Some properties of hyperspherical harmonics.  Journal of Mathematical Physics, 26(3):396–403, 1985.