Binomial and logarithmic polyharmonic expansions and addition theorems

Gegenbauer expansions and addition theorems for a binomial and logarithmic
fundamental solution of the even-dimensional Euclidean polyharmonic equation

Howard S. Cohl∗, Jessie E. Hirtenstein†, Jim Lawrence ‡§ and Lisa Ritter §

H. S. Cohl, J. E. Hirtenstein, J. Lawrence, L. Ritter

† Applied and Computational Mathematics Division, National Institute of
Standards and Technology, Mission Viejo, CA 92694, USA
http://www.nist.gov/itl/math/msg/howard-s-cohl.cfm<EMAIL_ADDRESS>
† Department of Physics, University of California, Davis, CA 95616, USA
<EMAIL_ADDRESS>
‡ Department of Mathematical Sciences, George Mason University, Fairfax, VA
22030, USA<EMAIL_ADDRESS>
§ Applied and Computational Mathematics Division, National Institute of
Standards and Technology, Gaithersburg, MD 20878, USA<EMAIL_ADDRESS>
Received ???, in final form ????; Published online ????

On even-dimensional Euclidean space for integer powers of the Laplace operator
greater than or equal to half the dimension, a fundamental solution of the
polyharmonic equation has binomial and logarithmic behavior. Gegenbauer
polynomial expansions of these fundamental solutions are obtained through a
limit applied to Gegenbauer expansions of a power-law fundamental solution of
the polyharmonic equation. This limit is accomplished through parameter
differentiation. By combining these results with previously derived azimuthal
Fourier series expansions for these binomial and logarithmic fundamental
solutions, we are able to obtain addition theorems for the azimuthal Fourier
coefficients. These logarithmic and binomial addition theorems are expressed
in Vilenkin polyspherical geodesic polar coordinate systems and as well in
generalized Hopf coordinates on spheres in arbitrary even dimensions.

logarithmic fundamental solutions; even dimensional Cartesian space;
Gegenbauer polynomial expansions

35A08, 31B30, 31C12, 33C05, 42A16

##  1 Introduction

Analysis of the polyharmonic operators (natural powers of the Laplace
operator) are ubiquitous in many areas of pure and applied mathematics and as
well in physics and engineering problems. Here we concern ourselves with a
fundamental solution of the polyharmonic equation (Laplace, biharmonic, etc.),
which is connected to solutions of the inhomogeneous polyharmonic equation.
Solutions to inhomogeneous polyharmonic equations are useful in many physical
applications including those areas related to Poisson’s equation such as
Newtonian gravity, electrostatics, magnetostatics, quantum direct and exchange
interactions [7, §1], etc. Furthermore, applications of higher-powers of the
Laplace operator include such varied areas as minimal surfaces [18], continuum
mechanics [14], mesh deformation [12], elasticity [15], Stokes flow [13],
geometric design [24], cubature formulae [21], mean value theorems (cf.
Pizzetti’s formula) [19], and Hartree-Fock calculations of nuclei [25].

Closed-form expressions for the Fourier expansions of a logarithmic
fundamental solution for the polyharmonic equation are extremely useful when
solving inhomogeneous polyharmonic problems on even-dimensional Euclidean
space, especially when a degree of rotational symmetry is involved. A
fundamental solution of the polyharmonic equation on $d$-dimensional Euclidean
space ${\mathbb{R}}^{d}$ has two arguments and therefore maps from a
$2d$-dimensional space to the reals. Solutions to the inhomogeneous
polyharmonic equation can be obtained by convolution of a fundamental solution
with an integrable source distribution. Eigenfunction decompositions of a
fundamental solution reduces the dimension of the resulting convolution
integral to obtain Dirichlet boundary values in order to solve the resulting
elliptic system, replacing it instead by a sum or an integral over some
parameter space. By taking advantage of rotational or nearly rotational
symmetry in the azimuthal Fourier decomposition of the source distribution,
one reduces the dimensionality of the resulting convolution integral and
obtains a rapidly convergent Fourier cosine expansion. In the case of an
axisymmetric (constant angular dependence) source distribution, the entire
contribution to the boundary values are determined by a single term in the
azimuthal Fourier series. These kinds of expansions have been previously shown
to be extremely effective in solving inhomogeneous problems (see for instance
the discussion [8] and those papers which cite it).

It is well-known (see for [20, p. 45], [11, p. 202]) that a fundamental
solution of the polyharmonic equation on $d$-dimensional Euclidean space
${\mathbb{R}}^{d}$ is given by combinations of power-law and logarithmic
functions of the global distance between two points. In [7], the Fourier
coefficients of a power-law fundamental solution of the polyharmonic equation
were obtained. Gegenbauer and Jacobi polynomial expansions of a power-law
fundamental solution of the polyharmonic equation (which generalize the
Fourier expansions presented in [7]) were obtained in [6]. The coefficients of
these expansions are seen to be given in terms of associated Legendre
functions. Fourier expansions of a logarithmic and binomial fundamental
solution of the polyharmonic equation were obtained in [5].

The work presented in this manuscript is concerned with computing the
Gegenbauer coefficients of binomial and logarithmic fundamental solutions of
the polyharmonic equation. One obtains a logarithmic fundamental solution for
the polyharmonic equation only on even-dimensional Euclidean space and only
when the power of the Laplace operator is greater than or equal to the
dimension divided by two. The most familiar example of a logarithmic
fundamental solution of the polyharmonic equation occurs in two-dimensions,
for a single-power of the Laplace operator, i.e., Laplace’s equation. In this
manuscript we present an approach for obtaining the Gegenbauer expansion of
binomial and logarithmic fundamental solutions of the polyharmonic equation by
parameter differentiation of a power-law fundamental solution of the
polyharmonic equation.

This manuscript is organized as follows. In Section 2 we introduce the
fundamental mathematical sets, sequences, functions and orthogonal
polynomials, which are necessary to understand the mathematical details of
this manuscript. In Section 3 we describe the properties of binomial and
logarithmic fundamental solutions of the polyharmonic equation in even-
dimensional Euclidean space ${\mathbb{R}}^{d}$—and in particular in
rotationally-invariant coordinate systems and in Vilenkin polyspherical
coordinates. In Section 4, we derive using a limit-derivative approach, the
Fourier cosine and Gegenbauer polynomial expansions for the kernels which
occur in binomial and logarithmic fundamental solutions of the polyharmonic
equation in Euclidean space. These kernels are $(z-x)^{p}$ and
$(z-x)^{p}\log(z-x)$, where $z>1$, $x\in(-1,1)$ and $p\in{\mathbb{N}}_{0}$. In
Section 5 we use the results presented in the previous sections to obtain the
azimuthal Fourier and Gegenbauer expansions for binomial and logarithmic
fundamental solution of the polyharmonic equation in rotationally-invariant
coordinate systems in even-dimensional Euclidean space. In Section 7 we derive
addition theorems for the azimuthal Fourier coefficients for the binomial and
logarithmic fundamental solutions of the polyharmonic equation in of even-
dimensional Euclidean space.

##  2 Preliminaries

Here we introduce some nomenclature which we will rely upon in the text below.
Throughout this paper we adopt the following set notations:
$\mathbb{N}_{0}:=\\{0\\}\cup\mathbb{N}=\\{0,1,2,3,\ldots\\}$, and we use the
sets ${\mathbb{Z}}$, ${\mathbb{R}}$, $\mathbb{C}$ to represent the integers,
real and complex numbers respectively. As is the common convention for
associated Legendre functions [1, (8.1.1)], for any expression of the form
$(z^{2}-1)^{\alpha}$, read this as
$(z^{2}-1)^{\alpha}:=(z+1)^{\alpha}(z-1)^{\alpha}$, for any fixed
$\alpha\in\mathbb{C}$ and $z\in\mathbb{C}\setminus(-\infty,1]$. Over the set
of complex numbers, we assume that empty sum vanishes and the empty product is
unity. Given two numbers: $r,r^{\prime}\in{\mathbb{R}}$, define

$r_{\lessgtr}:=\begin{subarray}{c}\text{min}\\\
\text{max}\end{subarray}\\{r,r^{\prime}\\}.$ (2.1)

The harmonic number $H_{n}\in{\mathbb{Q}}$, $n\in{\mathbb{Z}}$, given by

$H_{n}:=\left\\{\begin{array}[]{ll}0,&\mbox{if}\ n\leq 0,\\\\[5.69046pt]
{\displaystyle\sum_{k=1}^{n}\frac{1}{k}},&\mbox{if}\ n\geq
1.\end{array}\right.$ (2.2)

The harmonic number is related to the polygamma function [9, (5.2.2)] with
$n\in{\mathbb{N}}_{0}$ since [16, p. 14]

$\psi(n+1)=-\gamma+H_{n},$

where $\gamma\approx 0.57721\ldots$ is the Euler-Mascheroni constant [9,
(5.2.3)].

The Pochhammer symbol is defined for $a\in\mathbb{C}$, $n\in{\mathbb{N}}$,
such that

$(a)_{0}:=1,\quad(a)_{n}:=(a)(a+1)\cdots(a+n-1).$ (2.3)

One also has

$(-p)_{n}=\left\\{\begin{array}[]{cl}{\displaystyle\frac{(-1)^{p}p!}{(p-n)!}}&\mbox{if}\
0\leq n\leq p,\\\\[11.38092pt] 0&\mbox{if}\ n\geq p+1,\end{array}\right.$
(2.4)

Note for $z\in{\mathbb{C}}$, $k\in{\mathbb{N}}_{0}$, the binomial coefficient
can be given in terms of the Pochhammer symbol as follows [9, (1.2.6)]

$\binom{z}{k}=\frac{(-1)^{k}(-z)_{k}}{k!}.$ (2.5)

The Gauss hypergeometric function
${}_{2}F_{1}:{\mathbb{C}}\times{\mathbb{C}}\times({\mathbb{C}}\setminus-{\mathbb{N}}_{0})\times{\mathbb{C}}\setminus[1,\infty)$
[9, Chapter 15] is defined as

${}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};z\right)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!},$

for $|z|<1$ and through analytical continuation for the rest of its domain. If
one takes $b=c$ in the Gauss hypergeometric function, one produces the
binomial theorem [9, (15.4.6)]

${}_{1}F_{0}\\!\left(\genfrac{}{}{0.0pt}{}{a}{-};z\right)=(1-z)^{-a}.$ (2.6)

Associated Legendre functions of the first and second kind
$P_{\nu}^{\mu},Q_{\nu}^{\mu}:{\mathbb{C}}\setminus(-\infty,1]\to{\mathbb{C}}$
[9, (14.3.6-7) and Section 14.21] are defined as

$P_{\nu}^{\mu}(z):=\frac{1}{\Gamma(1-\mu)}\left(\frac{z+1}{z-1}\right)^{\frac{\mu}{2}}{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{-\nu,\nu+1}{1-\mu};\frac{1-z}{2}\right),$
(2.7)

for $|1-z|<2$,

$Q_{\nu}^{\mu}(z):=\frac{\sqrt{\pi}{\mathrm{e}}^{i\pi\mu}\Gamma(\nu+\mu+1)(z^{2}-1)^{\frac{\mu}{2}}}{2^{\nu+1}\Gamma(\nu+\frac{3}{2})z^{\nu+\mu+1}}{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{\frac{\nu+\mu+1}{2},\frac{\nu+\mu+2}{2}}{\nu+\frac{3}{2}};\frac{1}{z^{2}}\right),$
(2.8)

for $|z|>1$, and elsewhere in $z$ by analytic continuation of the Gauss
hypergeometric function. A property of the associated Legendre functions that
we will take advantage of are the Whipple formulae [9, (14.9.16-17)] (these
are equivalent to each other)

$\displaystyle
Q_{\nu}^{\mu}(z)=\sqrt{\frac{\pi}{2}}{\mathrm{e}}^{i\mu\pi}\Gamma(\nu+\mu+1)(z^{2}-1)^{-\frac{1}{4}}P_{-\mu-\frac{1}{2}}^{-\nu-\frac{1}{2}}\left(\frac{z}{\sqrt{z^{2}-1}}\right),$
(2.9) $\displaystyle
P_{\nu}^{\mu}(z)=i\sqrt{\frac{2}{\pi}}\frac{{\mathrm{e}}^{i\nu\pi}(z^{2}-1)^{-\frac{1}{4}}}{\Gamma(-\nu-\mu)}Q_{-\mu-\frac{1}{2}}^{-\nu-\frac{1}{2}}\left(\frac{z}{\sqrt{z^{2}-1}}\right),$
(2.10)

for $\Re z>0$. These allow one to convert between the associated Legendre
functions of the first and second kind. We also take advantage of [9,
(14.9.14)]

$Q_{\nu}^{-\mu}(z)={\mathrm{e}}^{-2i\pi\mu}\frac{\Gamma(\nu-\mu+1)}{\Gamma(\nu+\mu+1)}Q_{\nu}^{\mu}(z).$
(2.11)

Some useful special cases are (cf. [9, (14.5.17)])

$Q_{\frac{1}{2}}^{\frac{1}{2}}(z)=\frac{i\sqrt{\pi/2}}{(z^{2}-1)^{\frac{1}{4}}(z+\sqrt{z^{2}-1})},$
(2.12)

and

$Q_{n+\frac{1}{2}}^{\frac{3}{2}}(z)=\frac{-i\sqrt{\pi/2}(z+(n+1)\sqrt{z^{2}-1})}{(z^{2}-1)^{\frac{3}{4}}(z+\sqrt{z^{2}-1})^{n+1}},$
(2.13)

which follows from [9, (14.10.6)] and [9, (14.5.17)]. One also has

$Q_{\mu-\frac{1}{2}}^{\mu+\frac{1}{2}}(z)=\frac{{\mathrm{e}}^{i\pi(\mu+\frac{1}{2})}2^{\mu-\frac{1}{2}}\Gamma(\mu+\frac{1}{2})}{(z^{2}-1)^{\frac{\mu}{2}+\frac{1}{4}}},$
(2.14)

where in this formula
$\mu\in{\mathbb{C}}\setminus\\{-\frac{1}{2},-1,-\frac{3}{2},\ldots\\}$, which
follows using (2.8) and the binomial theorem (2.6). Let
$p,m\in{\mathbb{N}}_{0}$.

In regard to parameter differentiation of associated Legendre functions of the
first kind, then Szmytkowski derived [23, cf. (5.12)]

$\displaystyle\left[\frac{\partial}{\partial\nu}P_{\nu}^{m}(z)\right]_{\nu=p}$
$\displaystyle=\left\\{\begin{array}[]{ll}{\displaystyle\left(\log\frac{z+1}{2}+2H_{2p}-H_{p}-H_{p-m}\right)P_{p}^{m}(z)}&\\\\[11.38092pt]
\hskip
14.22636pt{}+{\displaystyle(-1)^{p+m}\sum_{k=0}^{p-m-1}(-1)^{k}\frac{(2k+2m+1)\left[1+\frac{k!(p+m)!}{(k+2m)!(p-m)!}\right]}{(p-m-k)(p+m+k+1)}P_{k+m}^{m}(z)}&\\\\[11.38092pt]
\hskip
14.22636pt{}+{\displaystyle(-1)^{p}\frac{(p+m)!}{(p-m)!}\sum_{k=0}^{m-1}\frac{(-1)^{k}(2k+1)}{(p-k)(p+k+1)}P_{k}^{-m}(z)},&\mbox{if}\
0\leq m\leq p,\\\\[17.07182pt]
(-1)^{p+m+1}(p+m)!(m-p-1)!P_{p}^{-m}(z),&\mbox{if}\ m\geq
p+1.\end{array}\right.$ (2.19)

One interesting consequence of Szmytkowski’s formula is the following
corollary which does not seem to have appeared elsewhere in the literature.

######  Corollary 2.1.

Let $p,m\in{\mathbb{N}}_{0}$, $0\leq m\leq p$,
$z\in{\mathbb{C}}\setminus(-\infty,1]$. Then

$\displaystyle\left[\frac{\partial}{\partial\nu}Q_{m-\frac{1}{2}}^{\nu+\frac{1}{2}}(z)\right]_{\nu=p}=\left[2H_{2p}-H_{p}-\gamma+i\pi+\log\frac{z+\sqrt{z^{2}-1}}{2\sqrt{z^{2}-1}}\right]Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(z)$
$\displaystyle\hskip
28.45274pt+(p-m)!\sum_{k=0}^{p-m-1}\frac{(2k+2m+1)\left[1+\frac{k!(p+m)!}{(k+2m)!(p-m)!}\right]}{k!(p-m-k)(p+m+k+1)}Q_{m-\frac{1}{2}}^{k+m+\frac{1}{2}}(z)$
$\displaystyle\hskip
28.45274pt+(p+m)!\sum_{k=0}^{m-1}\frac{2k+1}{(k+m)!(p-k)(p+k+1)}Q_{m-\frac{1}{2}}^{k+\frac{1}{2}}(z).$
(2.20)

###### Proof.

Starting with (2.10) for $\mu=m\in{\mathbb{N}}_{0}$, and the reflection
formula for the gamma function [9, (5.5.3)], we have

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

Then replace

$\displaystyle\left[\frac{\partial}{\partial\nu}P_{\nu}^{m}(z)\right]_{\nu=p}=\frac{-i}{(z^{2}-1)^{\frac{1}{4}}}\sqrt{\frac{2}{\pi}}\left[\frac{\partial}{\partial\nu}\frac{{\mathrm{e}}^{-i\pi\nu}}{\Gamma(\nu-m+1)}Q_{m-\frac{1}{2}}^{\nu+\frac{1}{2}}\left(\frac{z}{\sqrt{z^{2}-1}}\right)\right]_{\nu=p}$
$\displaystyle=\frac{-i}{(z^{2}-1)^{\frac{1}{4}}}\sqrt{\frac{2}{\pi}}\left\\{\left[\frac{\partial}{\partial\nu}\frac{{\mathrm{e}}^{-i\pi\nu}}{\Gamma(\nu-m+1)}\right]_{\nu=p}Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}\left(\frac{z}{\sqrt{z^{2}-1}}\right)+\frac{(-1)^{p}}{(p-m)!}\left[\frac{\partial}{\partial\nu}Q_{m-\frac{1}{2}}^{\nu+\frac{1}{2}}\left(\frac{z}{\sqrt{z^{2}-1}}\right)\right]_{\nu=p}\right\\},$

with

$\left[\frac{\partial}{\partial\nu}\frac{{\mathrm{e}}^{-i\pi\nu}}{\Gamma(\nu-m+1)}\right]_{\nu=p}=\frac{(-1)^{p+1}}{(p-m)!}\left(i\pi+\psi(p-m+1)\right),$

in (2.19) for $0\leq m\leq p$, together with (2.10), [9, (14.9.14)], and
applying the map $z\mapsto\frac{z}{\sqrt{z^{2}-1}}$ [4, Appendix A], completes
the proof. Unfortunately, applying this same method for $m\geq p+1$ does not
seem to produce an analogous result. ∎

On the other hand, using the map $z\mapsto\frac{z}{\sqrt{z^{2}-1}}$, (2.19),
and (2.9), produces the following useful relation

$\displaystyle\left[\frac{\partial}{\partial\nu}P_{\nu}^{m}\left(\frac{z}{\sqrt{z^{2}-1}}\right)\right]_{\nu=p}=i\sqrt{\frac{2}{\pi}}(z^{2}-1)^{\frac{1}{4}}$
$\displaystyle\hskip
11.38092pt\times\left\\{\begin{array}[]{ll}{\displaystyle\frac{(-1)^{p+1}}{(p-m)!}\left(\log\frac{z+\sqrt{z^{2}-1}}{2\sqrt{z^{2}-1}}+2H_{2p}-H_{p}-H_{p-m}\right)Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(z)}&\\\\[11.38092pt]
\hskip
14.22636pt{}+{\displaystyle(-1)^{p+1}\sum_{k=0}^{p-m-1}\frac{(2k+2m+1)}{k!(p-m-k)(p+m+k+1)}\left[1+\frac{k!(p+m)!}{(k+2m)!(p-m)!}\right]Q_{m-\frac{1}{2}}^{k+m+\frac{1}{2}}(z)}&\\\\[11.38092pt]
\hskip
14.22636pt{}+{\displaystyle\frac{(-1)^{p+1}(p+m)!}{(p-m)!}\sum_{k=0}^{m-1}\frac{2k+1}{(k+m)!(p-k)(p+k+1)}Q_{m-\frac{1}{2}}^{k+\frac{1}{2}}(z)},&\mbox{if}\
0\leq m\leq p,\\\\[17.07182pt] (-1)^{m}(m-p-1)!\
Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(z),&\mbox{if}\ m\geq p+1.\end{array}\right.$
(2.25)

One main orthogonal polynomial that we use in this manuscript is the
Gegenbauer polynomial. The Gegenbauer polynomial
$C_{n}^{\nu}:{\mathbb{C}}\to{\mathbb{C}}$ can be defined by

$C_{n}^{\nu}(x):=\frac{(2\nu)_{n}}{n!}{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{-n,n+2\nu}{\nu+\frac{1}{2}};\frac{1-x}{2}\right),$
(2.26)

where $n\in{\mathbb{N}}_{0}$, $\nu\in(-\frac{1}{2},\infty)\setminus\\{0\\}.$
The Gegenbauer polynomial can also be written in terms of the Ferrers function
of the first kind (associated Legendre function of the first kind on-the-cut)
[9, (18.11.1)]

$\displaystyle
C_{n-m}^{m+\frac{1}{2}}(x)=\frac{1}{(\frac{1}{2})_{m}(-2)^{m}(1-x^{2})^{\frac{m}{2}}}{\sf
P}_{n}^{m}(x),$ (2.27)

where $x\in(-1,1)$, $n,m\in{\mathbb{N}}_{0}$ such that $0\leq m\leq n$. The
Ferrers function of the first kind is defined as [9, (14.3.1)]

${\sf
P}_{\nu}^{\mu}(x)=\frac{1}{\Gamma(1-\mu)}\left(\frac{1+x}{1-x}\right)^{\frac{1}{2}\mu}{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{-\nu,\nu+1}{1-\mu};\frac{1-x}{2}\right),$
(2.28)

where $\nu,\mu\in{\mathbb{C}}$,
$x\in{\mathbb{C}}\setminus((-\infty,-1]\cup[1,\infty))$. One may consider the
limit as $\mu\to 0$ of the Gegenbauer polynomial [2, (6.4.13)]

$\lim_{\mu\to 0}\frac{{n}+\mu}{\mu}C_{n}^{\mu}(x)=\epsilon_{n}T_{n}(x)$ (2.29)

where $\epsilon_{n}=2-\delta_{{n},0}$ is the Neumann factor, commonly
appearing in Fourier cosine series. The Chebyshev polynomial of the first kind
$T_{n}:{\mathbb{C}}\to{\mathbb{C}},$ is defined by

$T_{n}(x):={}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{-n,n}{\frac{1}{2}};\frac{1-x}{2}\right),$

with the useful Fourier cosine representation
$T_{n}(\cos\theta)=\cos(n\theta).$ The Chebyshev polynomial of the second kind
$U_{n}:{\mathbb{C}}\to{\mathbb{C}}$, is defined by [9, (18.7.4)]
$U_{n}(x):=C_{n}^{1}(x).$

For generalized Hopf coordinates (see Section 6.2 below) we use Jacobi
polynomials which are defined as [9, (18.5.7)]

$P_{n}^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_{n}}{n!}{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{-n,\alpha+\beta+n+1}{\alpha+1};\frac{1-z}{2}\right).$

######  Theorem 2.2.

Let $\alpha,\beta,n\in{\mathbb{N}}_{0}$, $z\in{\mathbb{C}}$. Then the Jacobi
polynomials have the following symmetry in parameter relations

$\displaystyle
P_{n}^{(\alpha,\beta)}(z)=\frac{(\alpha+n)!(\beta+n)!}{n!(\alpha+\beta+n)!}\left(\frac{z-1}{2}\right)^{-\alpha}P_{\alpha+n}^{(-\alpha,\beta)}(z),$
(2.30) $\displaystyle
P_{n}^{(\alpha,\beta)}(z)=\frac{(\alpha+n)!(\beta+n)!}{(\alpha+\beta+n)!n!}\left(\frac{z+1}{2}\right)^{-\beta}P_{\beta+n}^{(\alpha,-\beta)}(z),$
(2.31)

###### Proof.

For (2.30), start with [16, p. 212]

$P_{n}^{(\alpha,\beta)}(z)=\binom{n+\beta}{n}\left(\frac{z-1}{2}\right)^{n}{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{-n,-n-\alpha}{\beta+1};\frac{z+1}{z-1}\right),$

replace $\alpha\mapsto-\alpha$, then $n\mapsto n+\alpha$. Then using (2.5),
and the reflection formula [9, (5.5.3)], the final result follows since for
$x,y\in{\mathbb{Z}}$, one has

$\lim_{\epsilon\to
0}\frac{\sin(\pi(x+y+2\epsilon))}{\sin(\pi(x+\epsilon))}=\cos(\pi
y)=(-1)^{y}.$

For (2.31) use [9, (18.5.8)]

$P_{n}^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_{n}}{n!}\left(\frac{z+1}{2}\right)^{n}{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{-n,-n-\beta}{\alpha+1};\frac{z-1}{z+1}\right),$

and replace first $\beta\mapsto-\beta$, and then $n\mapsto n+\beta$. ∎

######  Corollary 2.3.

Let $\alpha,\beta\in{\mathbb{Z}}$, $n\in{\mathbb{N}}_{0}$, $z\in{\mathbb{C}}$.
Then the Jacobi polynomials have the following symmetry in parameter relation

$\displaystyle
P_{n}^{(\alpha,\beta)}(z)=\left(\frac{z-1}{2}\right)^{-\alpha}\left(\frac{z+1}{2}\right)^{-\beta}P_{\alpha+\beta+n}^{(-\alpha,-\beta)}(z).$
(2.32)

###### Proof.

For (2.32) use (2.30) and then substitute (2.31) to replace the Jacobi
polynomial which appears on the right-hand side. Notice that the restriction
$\alpha,\beta\in{\mathbb{N}}_{0}$ is relaxed to $\alpha,\beta\in{\mathbb{Z}}$
in the resulting expression. ∎

##  3 Binomial and logarithmic kernels for the even dimensional fundamental
solution of the polyharmonic equation in Euclidean space

If ${\bf x},{{\bf x}^{\prime}}\in{\mathbb{R}}^{d}$ then the Euclidean inner
product $(\cdot,\cdot):{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$
defined by

$({\bf x},{{\bf
x}^{\prime}}):=x_{1}x_{1}^{\prime}+\cdots+x_{d}x_{d}^{\prime},$ (3.1)

induces a norm (the Euclidean norm) $\|\cdot\|:{\mathbb{R}}^{d}\to[0,\infty)$,
on ${\mathbb{R}}^{d}$, given by $\|{\bf x}\|:=\sqrt{({\bf x},{{\bf
x}^{\prime}})}$. If $\Phi:{\mathbb{R}}^{d}\to{\mathbb{R}}$ satisfies the
polyharmonic equation given by

$(-\Delta)^{k}\Phi({\bf x})=0,$ (3.2)

where $k\in{\mathbb{N}}$ and $\Phi\in C^{2k}({\mathbb{R}}^{d}),$ ${\bf
x}\in{\mathbb{R}}^{d},$ and $\Delta:C^{p}({\mathbb{R}}^{d})\to
C^{p-2}({\mathbb{R}}^{d})$ for $p\geq 2$, is the Laplace operator on
$\mathbb{R}^{d}$ defined by $\Delta:=\frac{\partial^{2}}{\partial
x_{1}^{2}}+\cdots+\frac{\partial^{2}}{\partial x_{d}^{2}}.$ Then $\Phi$ is
referred to as polyharmonic, and $(-\Delta)^{k}$ is referred to as the
polyharmonic operator. If the power $k$ of the Laplace operator equals two,
then (3.2) is referred to as the biharmonic equation and $\Phi$ is called
biharmonic. The inhomogeneous polyharmonic equation is given by

$(-\Delta)^{k}\Phi({\bf x})=\rho({\bf x}),$ (3.3)

where we take $\rho$ to be an integrable function so that a solution to (3.3)
exists. A fundamental solution for the polyharmonic equation on
${\mathbb{R}}^{d}$ is a function
${{\mathcal{G}}}_{k}^{d}:({\mathbb{R}}^{d}\times{\mathbb{R}}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbb{R}}^{d}\\}\to{\mathbb{R}}$ which satisfies the
distributional equation

$(-\Delta)^{k}{{\mathcal{G}}}_{k}^{d}({\bf x},{\bf x}^{\prime})=\delta({\bf
x}-{\bf x}^{\prime}),$ (3.4)

where $\delta$ is the Dirac delta distribution and ${{\bf
x}^{\prime}}\in{\mathbb{R}}^{d}$.

Let $n\in{\mathbb{N}}$ and define the harmonic number
$H_{n}:=\sum_{k=1}^{n}\frac{1}{k}$. Then a fundamental solution of the
polyharmonic equation (3.2) on Euclidean space ${\mathbb{R}}^{d}$ is given by
(see for instance [3, (2.1)], [5, Theorem 1], [21, Section II.2])

${\mathcal{G}}_{k}^{d}({\bf x},{\bf
x}^{\prime}):=\left\\{\begin{array}[]{ll}{\displaystyle\frac{(-1)^{k+\frac{d}{2}+1}\,\|{\bf
x}-{\bf
x}^{\prime}\|^{2k-d}}{(k-1)!\,\left(k-\frac{d}{2}\right)!\,2^{2k-1}\pi^{\frac{d}{2}}}\left(\log\|{\bf
x}-{\bf x}^{\prime}\|-\beta_{k-\frac{d}{2},d}\right),}\hskip
18.75026pt\mathrm{if}\ d\,\,\mathrm{even},\ k\geq\frac{d}{2},\\\\[10.0pt]
{\displaystyle\frac{\Gamma(\frac{d}{2}-k)\|{\bf x}-{\bf
x}^{\prime}\|^{2k-d}}{(k-1)!\,2^{2k}\pi^{\frac{d}{2}}}}\hskip
165.59525pt\mathrm{otherwise},\end{array}\right.$ (3.5)

where $\beta_{p,d}\in{\mathbb{Q}}$ is defined as
$\beta_{p,d}:=\frac{1}{2}(H_{p}+H_{\frac{d}{2}+p-1}-H_{\frac{d}{2}-1}).$

######  Remark 3.1.

In regard to a logarithmic fundamental solution of the polyharmonic equation
($d$ even, $k\geq\frac{d}{2}$), note that [21, Section II.2] is missing the
term proportional to $\|{\bf x}-{{\bf x}^{\prime}}\|^{2k-d}$. This term is in
the kernel of the polyharmonic operator $(-\Delta)^{k}$, so for any constant
multiple of this term $\beta_{p,d}$ may be added to a fundamental solution of
the polyharmonic equation. Our choice for this constant is given so that

$-\Delta{\mathcal{G}}_{k}^{d}={\mathcal{G}}_{k-1}^{d},$ (3.6)

is satisfied for all $k>\frac{d}{2}$, and that for $k=\frac{d}{2}$, the
constant vanishes. Boyling’s fundamental solution satisfies (3.6) for all
$k>\frac{d}{2}$, but is missing the term proportional to $H_{\frac{d}{2}-1}$,
and therefore only vanishes when $k=\frac{d}{2}$ for $d=2$. Sobolev does not
include this constant term, so for him ${\mathcal{G}}_{k}^{d}$ is purely
logarithmic for all $k\geq\frac{d}{2}$, $d\geq 2$ even. However in that case
(3.6) is not satisfied for $k>\frac{d}{2}$.

We consider parametrizations of Euclidean space ${\mathbb{R}}^{d}$ which are
given in terms of coordinate systems whose coordinates are curvilinear, i.e.,
based on some transformation which converts the Cartesian coordinates to a
coordinate system with the same number of coordinates in which the coordinate
lines are curved. We consider solutions of the polyharmonic equation (3.2) in
a curvilinear coordinate system, which arises through the theory of separation
of variables. We refer to coordinate systems which yield solutions through the
separation of variables method as separable. In this manuscript, we restrict
our attention to separable rotationally-invariant coordinate systems for the
polyharmonic equation on ${\mathbb{R}}^{d}$ which are given by

$\left.\begin{array}[]{rcl}x_{1}&=&x_{1}(\xi_{1},\ldots,\xi_{d-1})\\\\[2.84544pt]
&\vdots&\\\\[2.84544pt]
x_{d-2}&=&x_{d-2}(\xi_{1},\ldots,\xi_{d-1})\\\\[2.84544pt]
x_{d-1}&=&R(\xi_{1},\ldots,\xi_{d-1})\cos\phi\\\\[2.84544pt]
x_{d}&=&R(\xi_{1},\ldots,\xi_{d-1})\sin\phi\\\\[2.84544pt]
\end{array}\quad\right\\}.$ (3.7)

These coordinate systems are described by $d$ coordinates: an angle
$\phi\in{\mathbb{R}}$ plus $(d-1)$-curvilinear coordinates
$(\xi_{1},\ldots,\xi_{d-1})$. Rotationally-invariant coordinate systems
parametrize points on the $(d-1)$-dimensional half-hyperplane given by
$\phi={\mathrm{const}}$ and $R\geq 0$ using the curvilinear coordinates
$(\xi_{1},\ldots,\xi_{d-1})$. A separable rotationally-invariant coordinate
system transforms the polyharmonic equation into a set of $d$-uncoupled
ordinary differential equations with separation constants $m\in{\mathbb{Z}}$
and $k_{j}$ for $1\leq j\leq d-2$. For a separable rotationally-invariant
coordinate system, this uncoupling is accomplished, in general, by assuming a
solution to (3.2) of the form

$\Phi({\bf
x})={\mathrm{e}}^{im\phi}\,{\mathcal{R}}(\xi_{1},\ldots,\xi_{d-1})\prod_{i=1}^{d-1}A_{i}(\xi_{i},m,k_{1},\ldots,k_{d-2}),$

where the properties of the functions ${\mathcal{R}}$ and $A_{i}$, for $1\leq
i\leq d-1$, and the constants $k_{j}$ for $1\leq j\leq d-2$, depend on the
specific separable rotationally-invariant coordinate system in question.
Separable coordinate systems are divided into two separate classes, those
which are simply separable (${\mathcal{R}}={\mathrm{const}}$), and those which
are ${\mathcal{R}}$-separable. For an extensive description of the theory of
separation of variables see [17].

The Euclidean distance between two points ${\bf x},{{\bf
x}^{\prime}}\in{\mathbb{R}}^{d}$, expressed in a rotationally-invariant
coordinate system, is given by

$\displaystyle\|{\bf x}-{{\bf
x}^{\prime}}\|=\sqrt{2RR^{\prime}}\left[\chi-\cos(\phi-\phi^{\prime})\right]^{\frac{1}{2}},$
(3.8)

where the hypertoroidal parameter $\chi>1$, is given by

$\chi:=\chi(R,R^{\prime},x_{1},\ldots,x_{d-2},x_{1}^{\prime},\ldots,x_{d-2}^{\prime}):=\frac{R^{2}+{R^{\prime}}^{2}+{\displaystyle\sum_{k=1}^{d-2}(x_{k}-x_{k}^{\prime})^{2}}}{\displaystyle
2RR^{\prime}},$ (3.9)

where $R,R^{\prime}\in(0,\infty)$ are defined in (3.7). The hypersurfaces
given by $\chi={\mathrm{const}}$ are independent of coordinate system and
represent hypertori of revolution.

One type of coordinate system which parametrizes points in $d$-dimensional
Euclidean space which has a high degree of symmetry are Vilenkin’s
polyspherical coordinates (for a detailed description of Vilenkin’s
polyspherical coordinates, see [6, Appendix B and references therein]). These
curvilinear orthogonal coordinate systems are composed of a radius
$r\in[0,\infty)$ and $(d-1)$ angles which must have domains given in
$\\{[0,\frac{1}{2}\pi],[0,\pi],[-\pi,\pi)\\}$. Using these coordinate systems,
we can also express the distance between two points as

$\|{\bf x}-{{\bf
x}^{\prime}}\|=\sqrt{2rr^{\prime}}(\zeta-\cos\gamma)^{\frac{1}{2}},$ (3.10)

where $\zeta:[0,\infty)^{2}\to[1,\infty)$ is defined by

$\zeta=\zeta(r,r^{\prime}):=\frac{r^{2}+r^{\prime 2}}{2rr^{\prime}},$ (3.11)

and the separation angle $\gamma\in[0,\pi]$ is defined through the relation

$\cos\gamma:=\frac{({\bf x},{{\bf x}^{\prime}})}{\|{\bf x}\|\|{{\bf
x}^{\prime}}\|}=\frac{({\bf x},{{\bf x}^{\prime}})}{rr^{\prime}},$ (3.12)

using the Euclidean inner product and norm. Note that

$\sqrt{\zeta^{2}-1}=\frac{r_{>}^{2}-r_{<}^{2}}{2rr^{\prime}}.$ (3.13)

From (3.5) we see that apart from a multiplicative constant, the algebraic
expression of a fundamental solution for the polyharmonic equation on
Euclidean space ${\mathbb{R}}^{d}$ for $d$ even, $k\geq\frac{d}{2},$ is given
by
${\mathfrak{l}}_{k}^{d}:({\mathbb{R}}^{d}\times{\mathbb{R}}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbb{R}}^{d}\\}\to{\mathbb{R}}$ defined by

${\mathfrak{l}}_{k}^{d}({\bf x},{{\bf x}^{\prime}}):=\|{\bf x}-{{\bf
x}^{\prime}}\|^{2k-d}\left(\log\|{\bf x}-{{\bf
x}^{\prime}}\|-\beta_{k-\frac{d}{2},d}\right).$ (3.14)

By expressing ${\mathfrak{l}}_{k}^{d}$ in a rotationally-invariant coordinate
system (3.7) one has the following result. Let $p=k-d/2\in{\mathbb{N}}_{0}$.
Then

$\displaystyle{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})=\left(2RR^{\prime}\right)^{p}\left(\frac{1}{2}\log\left(2RR^{\prime}\right)-\beta_{p,d}\right)\left(\chi-\cos(\phi-\phi^{\prime})\right)^{p}$
$\displaystyle\hskip
142.26378pt+\frac{1}{2}\left(2RR^{\prime}\right)^{p}\left(\chi-\cos(\phi-\phi^{\prime})\right)^{p}\log\left(\chi-\cos(\phi-\phi^{\prime})\right).$
(3.15)

Also, in a Vilenkin polyspherical coordinate system, one has

$\displaystyle{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})=\left(2rr^{\prime}\right)^{p}\left(\frac{1}{2}\log\left(2rr^{\prime}\right)-\beta_{p,d}\right)\left(\zeta-\cos\gamma\right)^{p}+\frac{1}{2}\left(2rr^{\prime}\right)^{p}\left(\zeta-\cos\gamma\right)^{p}\log\left(\zeta-\cos\gamma\right).$
(3.16)

For the polyharmonic equation on even-dimensional Euclidean space
${\mathbb{R}}^{d}$, if $1\leq k\leq\frac{d}{2}-1$ then a fundamental solution
is given by
${\mathfrak{h}}_{k}^{d}:({\mathbb{R}}^{d}\times{\mathbb{R}}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbb{R}}^{d}\\}\to{\mathbb{R}}$ which is a power-law
given by ${\mathfrak{h}}_{k}^{d}({\bf x},{{\bf x}^{\prime}})=\|{\bf x}-{{\bf
x}^{\prime}}\|^{2k-d}$, where $2k-d\in-2{\mathbb{N}}$. For this range of $k$
values then $2k-d$ is a negative even integer and this case and all its
implications are fully covered by the material presented in [6].

For the case in which a logarithmic fundamental solution exists, namely
$k\geq\frac{d}{2}$, then $2k-d\in 2{\mathbb{N}}$ is a positive even integer
and the kernel
${\mathfrak{j}}_{k}^{d}:{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$

${\mathfrak{j}}_{k}^{d}({\bf x},{{\bf x}^{\prime}}):=\|{\bf x}-{{\bf
x}^{\prime}}\|^{2k-d}.$ (3.17)

corresponds to a binomial expression $(z-x)^{p}$, where
$p=k-d/2\in{\mathbb{N}}_{0}$ and its series expansion either in terms of
Chebyshev polynomials of the first kind or in terms of Gegenbauer polynomials
truncates in a finite number of terms. By expressing ${\mathfrak{j}}_{k}^{d}$
in a rotationally-invariant coordinate system there is

${\mathfrak{j}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})=\left(2RR^{\prime}\right)^{p}\left[\chi-\cos(\phi-\phi^{\prime})\right]^{p},$
(3.18)

and expressing ${\mathfrak{j}}_{k}^{d}$ in a Vilenkin polyspherical coordinate
system one has

${\mathfrak{j}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})=\left(2rr^{\prime}\right)^{p}\left[\zeta-\cos\gamma\right]^{p},$
(3.19)

where $p=k-d/2\in{\mathbb{N}}_{0}$.

##  4 Fourier and Gegenbauer expansions of binomial and logarithmic kernels

We require Gegenbauer expansions for kernels which naturally arise in a
logarithmic fundamental solution of the polyharmonic equation in Vilenkin’s
polyspherical coordinates in even-dimensional Euclidean space. Since these
coordinates are rotationally invariant, in this study of addition theorems in
these coordinates, we require Fourier cosine (Chebyshev polynomials of the
first kind) and Gegenbauer polynomial expansions (see [6]). By necessity, here
we treat the binomial $(z-x)^{p}$ and logarithmic $(z-x)^{p}\log(z-x)$
kernels, where $x,z,\nu\in{\mathbb{C}}$ and $p\in{\mathbb{N}}_{0}$. We have
previously derived Fourier expansions of the binomial [7, (3.10)] and the
logarithmic [5, (20), (26)] kernels. We now treat those corresponding
Gegenbauer polynomial expansions. By examining (3.15), (3.16), (3.18), we see
that for the computation of Fourier and Gegenbauer expansions, we are
interested in the Fourier and Gegenbauer expansions of the Euler and
logarithmic kernels.

The formulas which are presented below all rely in one way or another on the
following important Gegenbauer polynomial expansion which can be found in [6,
(3.4)], namely

$(z-x)^{-\nu}=\frac{2^{\mu+\frac{1}{2}}\Gamma(\mu)}{\sqrt{\pi}\,\Gamma(\nu)}{\mathrm{e}}^{i\pi(\mu-\nu+\frac{1}{2})}(z^{2}-1)^{\frac{\mu-\nu}{2}+\frac{1}{4}}\sum_{n=0}^{\infty}(n+\mu)C_{n}^{\mu}(x)Q_{n+\mu-\frac{1}{2}}^{\nu-\mu-\frac{1}{2}}(z),$
(4.1)

where $z\in{\mathbb{C}}\setminus(-\infty,1]$, and $x\in{\mathbb{C}}$ lies
inside the ellipse with foci at $\pm 1$ that passes through $z$. This result
and the following results have the curious property of $z,x$ lying on ellipses
with foci at $\pm 1$ and this is due to the important theorem of Szegő, namely
[22, Theorem 12.7.3, Expansion of an analytic function in terms of orthogonal
polynomials].

######  Lemma 4.1.

Let $p\in{\mathbb{N}}_{0}$, $z\in{\mathbb{C}}\setminus(-\infty,1]$,
$x\in{\mathbb{C}}$. Then the expansion of Euler kernel in Chebyshev
polynomials of the first kind is given by the following binomial expansion
(see [7, (4.4)])

$\displaystyle(z-x)^{p}=i(-1)^{p+1}\sqrt{\frac{2}{\pi}}p!(z^{2}-1)^{\frac{p}{2}+\frac{1}{4}}\sum_{n=0}^{p}\frac{(-1)^{n}\epsilon_{n}T_{n}(x)}{(p-n)!(p+n)!}Q_{n-\frac{1}{2}}^{p+\frac{1}{2}}(z),$
(4.2)

where $p\in{\mathbb{N}}_{0}$.

###### Proof.

This truncated series result follows from (4.1) with
$-\nu=p\in{\mathbb{N}}_{0}$ and then taking $\mu\to 0$ with (2.29). ∎

The Fourier cosine expansion of the important logarithmic kernel is given in
the following lemma.

######  Lemma 4.2.

Let $p\in{\mathbb{N}}_{0}$, $z\in{\mathbb{C}}\setminus(-\infty,1]$,
$z-x\in{\mathbb{C}}\setminus(-\infty,0]$, $x$ lies inside the ellipse with
foci at $\pm 1$ that passes through $z$. Then

$\displaystyle(z-x)^{p}\log(z-x)=(z-x)^{p}\left(\log\frac{z+\sqrt{z^{2}-1}}{2}+2H_{2p}\right)$
$\displaystyle\hskip
5.69046pt+i\sqrt{\frac{2}{\pi}}(-1)^{p}p!\left(z^{2}-1\right)^{\frac{p}{2}+\frac{1}{4}}\sum_{n=0}^{p}\frac{\epsilon_{n}(-1)^{n}T_{n}(x)}{(p-n)!(p+n)!}\left(H_{p+n}+H_{p-n}\right)Q_{n-\frac{1}{2}}^{p+\frac{1}{2}}(z)$
$\displaystyle\hskip
5.69046pt+i\sqrt{\frac{2}{\pi}}(-1)^{p+1}p!\left(z^{2}-1\right)^{\frac{p}{2}+\frac{1}{4}}\sum_{n=0}^{p-1}\frac{\epsilon_{n}(-1)^{n}T_{n}(x)}{(p+n)!}\sum_{k=0}^{p-n-1}\frac{(2n+2k+1)\left[1+\frac{k!(p+n)!}{(2n+k)!(p-n)!}\right]}{k!(p-n-k)(p+n+k+1)}Q_{n-\frac{1}{2}}^{n+k+\frac{1}{2}}(z)$
$\displaystyle\hskip
5.69046pt+i\frac{2^{\frac{3}{2}}}{\sqrt{\pi}}(-1)^{p+1}p!\left(z^{2}-1\right)^{\frac{p}{2}+\frac{1}{4}}\sum_{n=1}^{p}\frac{(-1)^{n}T_{n}(x)}{(p-n)!}\sum_{k=0}^{n-1}\frac{2k+1}{(n+k)!(p-k)(p+k+1)}Q_{n-\frac{1}{2}}^{k+\frac{1}{2}}(z)$
$\displaystyle\hskip
5.69046pt+i\frac{2^{\frac{3}{2}}}{\sqrt{\pi}}p!\left(z^{2}-1\right)^{\frac{p}{2}+\frac{1}{4}}\sum_{n=p+1}^{\infty}\frac{(n-p-1)!\,T_{n}(x)}{(p+n)!}Q_{n-\frac{1}{2}}^{p+\frac{1}{2}}(z).$
(4.3)

###### Proof.

Use [5, (26)], with (2.9), (2.10), (2.11). ∎

The following consequence of (4.3) is also given in [16, p. 259].

######  Corollary 4.3.

Let $z\in{\mathbb{C}}\setminus(-\infty,1]$,
$z-x\in{\mathbb{C}}\setminus(-\infty,0]$, $x$ lies inside the ellipse with
foci at $\pm 1$ that passes through $z$. Then

$\log(z-x)=\log\frac{z+\sqrt{z^{2}-1}}{2}-2\sum_{n=1}^{\infty}\frac{T_{n}(x)}{n}\frac{1}{(z+\sqrt{z^{2}-1})^{n}}.$

###### Proof.

Let $p=0$ in (4.3) and use [9, (14.5.17)]. This completes the proof. ∎

The Gegenbauer polynomial expansion of the binomial is given in the following
lemma.

######  Lemma 4.4.

Let $p\in{\mathbb{N}}_{0}$, $\mu\in(-\frac{1}{2},\infty)\setminus\\{0\\}$,
$z\in{\mathbb{C}}\setminus(-\infty,1]$, $x\in{\mathbb{C}}$. Then

$(z-x)^{p}=\frac{2^{\mu+\frac{1}{2}}}{\sqrt{\pi}}{\mathrm{e}}^{i\pi(p-\mu-\frac{1}{2})}\Gamma(\mu)p!(z^{2}-1)^{\frac{p+\mu}{2}+\frac{1}{4}}\sum_{n=0}^{p}\frac{(-1)^{n}(n+\mu)C_{n}^{\mu}(x)}{(p-n)!\Gamma(n+p+2\mu+1)}Q_{n+\mu-\frac{1}{2}}^{p+\mu+\frac{1}{2}}(z).$
(4.4)

###### Proof.

Starting with [6, (3.4)], (4.1), we take the limit $\nu\to-p,$ producing

$(z-x)^{p}=\frac{2^{\mu+\frac{1}{2}}}{\sqrt{\pi}}{\mathrm{e}}^{i\pi(\mu+p+\frac{1}{2})}\Gamma(\mu)(z^{2}-1)^{\frac{\mu+p}{2}+\frac{1}{4}}\sum_{n=0}^{\infty}(n+\mu)C_{n}^{\mu}(x)\lim_{\nu\to-p}\frac{1}{\Gamma(\nu)}Q_{n+\mu-\frac{1}{2}}^{\nu-\mu-\frac{1}{2}}(z).$
(4.5)

By using (2.11), (2.4), we have

$\lim_{\nu\to-p}\frac{1}{\Gamma(\nu)}Q_{n+\mu-\frac{1}{2}}^{\nu-\mu-\frac{1}{2}}(z)=\left\\{\begin{array}[]{cl}{\displaystyle\frac{(-1)^{n+1}{\mathrm{e}}^{-2i\pi\mu}p!}{(p-n)!\Gamma(n+p+2\mu+1)}Q_{n+\mu-\frac{1}{2}}^{p+\mu+\frac{1}{2}}(z)}&\mbox{if}\
0\leq n\leq p,\\\\[11.38092pt] 0&\mbox{if}\ n\geq p+1.\end{array}\right.$

Using this limit in (4.5) completes the proof. ∎

Note that the above lemma generalizes (4.2) by the limit formula (2.29).

The Gegenbauer polynomial expansion of the logarithmic kernel is given as
follows.

######  Lemma 4.5.

Let $p\in{\mathbb{N}}_{0}$, $\mu\in{\mathbb{N}}$,
$z\in{\mathbb{C}}\setminus(-\infty,1]$,
$z-x\in{\mathbb{C}}\setminus(-\infty,0]$, $x$ lies inside the ellipse with
foci at $\pm 1$ that passes through $z$. Then

$\displaystyle(z-x)^{p}\log(z-x)=(z-x)^{p}\left(\log\frac{z+\sqrt{z^{2}-1}}{2}+2H_{2p+2\mu}+H_{p}-H_{p+\mu}\right)$
$\displaystyle\hskip
5.69046pt+i(-1)^{p+\mu}\frac{2^{\mu+\frac{1}{2}}}{\sqrt{\pi}}p!(\mu-1)!\left(z^{2}-1\right)^{\frac{p+\mu}{2}+\frac{1}{4}}\sum_{n=0}^{p}\frac{(n+\mu)(-1)^{n}C_{n}^{\mu}(x)}{(p-n)!(p+n+2\mu)!}\left(H_{p+n+2\mu}+H_{p-n}\right)Q_{n+\mu-\frac{1}{2}}^{p+\mu+\frac{1}{2}}(z)$
$\displaystyle\hskip
5.69046pt+i(-1)^{p+\mu+1}\frac{2^{\mu+\frac{1}{2}}}{\sqrt{\pi}}p!(\mu-1)!\left(z^{2}-1\right)^{\frac{p+\mu}{2}+\frac{1}{4}}\sum_{n=0}^{p-1}\frac{(n+\mu)(-1)^{n}C_{n}^{\mu}(x)}{(p+n+2\mu)!}$
$\displaystyle\hskip
34.14322pt\times\sum_{k=0}^{p-n-1}\frac{(2n+2k+2\mu+1)}{k!(p-n-k)(p+n+k+2\mu+1)}\left[1+\frac{k!(p+n+2\mu)!}{(k+2n+2\mu)!(p-n)!}\right]Q_{n+\mu-\frac{1}{2}}^{k+n+\mu+\frac{1}{2}}(z)$
$\displaystyle\hskip
5.69046pt+i(-1)^{p+\mu+1}\frac{2^{\mu+\frac{1}{2}}}{\sqrt{\pi}}p!(\mu-1)!\left(z^{2}-1\right)^{\frac{p+\mu}{2}+\frac{1}{4}}\sum_{n=0}^{p}\frac{(n+\mu)(-1)^{n}C_{n}^{\mu}(x)}{(p-n)!}$
$\displaystyle\hskip
34.14322pt\times\sum_{k=0}^{n+\mu-1}\frac{2k+1}{(n+k+\mu)!(p+\mu-k)(p+k+\mu+1)}Q_{n+\mu-\frac{1}{2}}^{k+\frac{1}{2}}(z)$
$\displaystyle\hskip
5.69046pt+i(-1)^{\mu}\frac{2^{\mu+\frac{1}{2}}}{\sqrt{\pi}}p!(\mu-1)!\left(z^{2}-1\right)^{\frac{p+\mu}{2}+\frac{1}{4}}\sum_{n=p+1}^{\infty}\frac{(n+\mu)(n-p-1)!C_{n}^{\mu}(x)}{(p+n+2\mu)!}Q_{n+\mu-\frac{1}{2}}^{p+\mu+\frac{1}{2}}(z).$
(4.6)

###### Proof.

Let $\nu\in{\mathbb{C}}$, $\mu\in(-\frac{1}{2},\infty)\setminus\\{0\\},$
$z\in{\mathbb{C}}\setminus(-\infty,1]$ and $x\in{\mathbb{C}}$ lies inside the
ellipse with foci at $\pm 1$ that passes through $z$. Using (4.1) and (2.9),
one has

$(z-x)^{\nu}=2^{\mu}\Gamma(\mu)(z^{2}-1)^{\frac{\nu+\mu}{2}}\sum_{n=0}^{\infty}(n+\mu)(-\nu)_{n}P_{\nu+\mu}^{-n-\mu}\left(\frac{z}{\sqrt{z^{2}-1}}\right)C_{n}^{\mu}(x).$
(4.7)

Let $p\in{\mathbb{N}}_{0}$. One then may use the following identity

$(z-x)^{p}\log(z-x)=\lim_{\nu\to 0}\frac{\partial}{\partial\nu}(z-x)^{\nu+p},$

which upon substituting $\nu\mapsto\nu+p$ in (4.7) produces

$(z-x)^{p}\log(z-x)=2^{\mu}(z^{2}-1)^{\frac{\mu+p}{2}}\sum_{n=0}^{\infty}(n+\mu)(-1)^{n}C_{n}^{\mu}(x)\lim_{\nu\to
0}\frac{\partial}{\partial\nu}\frac{(z^{2}-1)^{\frac{\nu}{2}}\Gamma(\nu+p+1)}{\Gamma(\nu+p+2\mu+1)}P_{\nu+p+\mu}^{n+\mu}\left(\frac{z}{\sqrt{z^{2}-1}}\right).$

Performing the derivatives, one obtains

$\displaystyle\frac{\partial}{\partial\nu}\frac{(z^{2}-1)^{\frac{\nu}{2}}\Gamma(\nu+p+1)}{\Gamma(\nu+p+2\mu+1)}P_{\nu+p+\mu}^{n+\mu}\left(\frac{z}{\sqrt{z^{2}-1}}\right)$
$\displaystyle=\frac{(z^{2}-1)^{\frac{\nu}{2}}\Gamma(\nu+p+1)}{\Gamma(\nu+p+2\mu+1)}\left(\log\sqrt{z^{2}-1}+\psi(p+\nu+1)-\psi(p+n+\nu+2\mu+1)\right)P_{\nu+p+\mu}^{n+\mu}\left(\frac{z}{\sqrt{z^{2}-1}}\right)$
$\displaystyle+\frac{(z^{2}-1)^{\frac{\nu}{2}}\Gamma(\nu+p+1)}{\Gamma(\nu+p+2\mu+1)}\frac{\partial}{\partial\nu}P_{\nu+p+\mu}^{n+\mu}\left(\frac{z}{\sqrt{z^{2}-1}}\right),$

and after taking the limit one has

$\displaystyle\lim_{\nu\to
0}\frac{\partial}{\partial\nu}\frac{p!}{\Gamma(p+2\mu+1)}P_{p+\mu}^{n+\mu}\left(\frac{z}{\sqrt{z^{2}-1}}\right)$
$\displaystyle=\frac{p!}{\Gamma(p+2\mu+1)}\left(\log\sqrt{z^{2}-1}+\psi(p+1)-\psi(p+n+2\mu+1)\right)P_{p+\mu}^{n+\mu}\left(\frac{z}{\sqrt{z^{2}-1}}\right)$
$\displaystyle+\frac{p!}{\Gamma(p+2\mu+1)}\left[\frac{\partial}{\partial\nu}P_{\nu}^{n+\mu}\left(\frac{z}{\sqrt{z^{2}-1}}\right)\right]_{\nu=p+\mu}.$

For general $\mu$, the parameter derivative of the associated Legendre
function of the first kind given in the above equation is not known. However
for $\mu\in{\mathbb{N}}_{0}$ it is known (see (2.19)). The expansion for the
logarithmic kernel for $\mu=0$ corresponds to the Chebyshev polynomial of the
first kind (see (2.29)), and therefore corresponds to (4.3). Hence from this
point forward we treat $\mu\in{\mathbb{N}}$, and the result follows using
(2.25). ∎

One interesting consequence of (4.6) is the following expansion.

######  Corollary 4.6.

Let $m\in{\mathbb{N}}$, $z\in{\mathbb{C}}\setminus(-\infty,1]$,
$z-x\in{\mathbb{C}}\setminus(-\infty,0]$, $x$ lies inside the ellipse with
foci at $\pm 1$ that passes through $z$. Then

$\displaystyle\log(z-x)=\log\frac{z+\sqrt{z^{2}-1}}{2}+H_{2m}-H_{m}$
$\displaystyle\hskip
56.9055pt+i(-1)^{m+1}\frac{2^{m+\frac{1}{2}}}{\sqrt{\pi}}m!(z^{2}-1)^{\frac{m}{2}+\frac{1}{4}}\sum_{k=0}^{m-1}\frac{2k+1}{(k+m)!(m-k)(k+m+1)}Q_{m-\frac{1}{2}}^{k+\frac{1}{2}}(z)$
$\displaystyle\hskip
56.9055pt+i(-1)^{m}\frac{2^{m+\frac{1}{2}}}{\sqrt{\pi}}(m-1)!(z^{2}-1)^{\frac{m}{2}+\frac{1}{4}}\sum_{n=1}^{\infty}\frac{(n+m)n!C_{n}^{m}(x)}{(2m+n)!}Q_{n+m-\frac{1}{2}}^{m+\frac{1}{2}}(z).$
(4.8)

###### Proof.

Let $p=0$ in (4.6) using the duplication theorem [9, (5.5.5)], and (2.14),
completes the proof. ∎

######  Corollary 4.7.

Let $z\in{\mathbb{C}}\setminus(-\infty,1]$,
$z-x\in{\mathbb{C}}\setminus(-\infty,0]$, $x$ lies inside the ellipse with
foci at $\pm 1$ that passes through $z$. Then

$\log(z-x)=\log\frac{z+\sqrt{z^{2}-1}}{2}+\frac{1}{2}-\frac{\sqrt{z^{2}-1}}{z+\sqrt{z^{2}-1}}-2\sum_{n=1}^{\infty}\frac{U_{n}(x)}{n(n+2)}\frac{z+(n+1)\sqrt{z^{2}-1}}{(z+\sqrt{z^{2}-1})^{n+1}}.$

###### Proof.

Let $\mu=1$ in (4.8), then use (2.12), (2.13). Simplification completes the
proof. ∎

##  5 Azimuthal Fourier and Gegenbauer polynomial expansions of binomial and
logarithmic fundamental solutions

The behavior of a logarithmic fundamental solution of the polyharmonic
equation on even-dimensional Euclidean space ${\mathbb{R}}^{d}$ in a
rotationally-invariant coordinate system and in a Vilenkin polyspherical
coordinate system are given respectively by (3.15), (3.16). For rotationally-
invariant coordinate systems (3.7), recall that $R,R^{\prime}$ are the
cylindrical radii and $\phi$, $\phi^{\prime}$ are azimuthal angles and that
$\chi$, the toroidal parameter, is defined in (3.9). Furthermore, for the
polyharmonic equation (3.2), $k\in{\mathbb{N}}$ is the power of the positive
Laplacian. For the definitions of the special functions and numbers used in
the results presented in this section, see Section 2. We now present the
expression for the azimuthal Fourier expansion of a logarithmic fundamental
solution of the polyharmonic equation.

######  Theorem 5.1.

Let ${\bf x},{{\bf x}^{\prime}}\in{\mathbb{R}}^{d}$ with $d\in 2{\mathbb{N}}$,
$p=k-d/2\in{\mathbb{N}}_{0}$. Then the azimuthal Fourier expansion of a
logarithmic fundamental solution of the polyharmonic equation
${\mathfrak{l}}_{k}^{d}$ in a rotationally-invariant coordinate system on
Euclidean space ${\mathbb{R}}^{d}$ is given by

$\displaystyle{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})=\frac{ip!}{\sqrt{2\pi}}\left(2RR^{\prime}\right)^{p}(\chi^{2}-1)^{\frac{p}{2}+\frac{1}{4}}\Biggl{[}\left(\log\left(RR^{\prime}\right)+\log(\chi+\sqrt{\chi^{2}-1})+2H_{2p}-H_{p}-H_{\frac{d}{2}+p-1}+H_{\frac{d}{2}-1}\right)$
$\displaystyle\hskip
28.45274pt\times(-1)^{p+1}\sum_{m=0}^{p}\frac{\epsilon_{m}(-1)^{m}\cos{\left(m(\phi-\phi^{\prime})\right)}}{(p-m)!(p+m)!}Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)$
$\displaystyle\hskip
5.69046pt+(-1)^{p}\sum_{m=0}^{p}\frac{\epsilon_{m}(-1)^{m}\cos{\left(m(\phi-\phi^{\prime})\right)}}{(p-m)!(p+m)!}\left(H_{p+m}+H_{p-m}\right)Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)$
$\displaystyle\hskip
5.69046pt+(-1)^{p+1}\sum_{m=0}^{p-1}\frac{\epsilon_{m}(-1)^{m}\cos{\left(m(\phi-\phi^{\prime})\right)}}{(p+m)!}\sum_{k=0}^{p-m-1}\frac{(2m+2k+1)\left[1+\frac{k!(p+m)!}{(2m+k)!(p-m)!}\right]}{k!(p-m-k)(p+m+k+1)}Q_{m-\frac{1}{2}}^{m+k+\frac{1}{2}}(\chi)$
$\displaystyle\hskip
5.69046pt+2(-1)^{p+1}\sum_{m=1}^{p}\frac{(-1)^{m}\cos{\left(m(\phi-\phi^{\prime})\right)}}{(p-m)!}\sum_{k=0}^{m-1}\frac{2k+1}{(m+k)!(p-k)(p+k+1)}Q_{m-\frac{1}{2}}^{k+\frac{1}{2}}(\chi)$
$\displaystyle\hskip
5.69046pt+2\sum_{m=p+1}^{\infty}\frac{(m-p-1)!\,\cos{\left(m(\phi-\phi^{\prime})\right)}}{(p+m)!}Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)\Biggr{]}.$
(5.1)

###### Proof.

Beginning with (3.15), applying the identities (4.2), (4.3) and simplifying,
completes the proof. ∎

A logarithmic fundamental solution of the polyharmonic equation expressed in a
Vilenkin polyspherical coordinate system is given by (3.16). Also recall that
$r$, $r^{\prime}$ are the hyperspherical radii and $\cos\gamma=({\bf x},{{\bf
x}^{\prime}})/(\|{\bf x}\|\|{{\bf x}^{\prime}}\|)$ (3.12) is the separation
angle (3.12) in a Vilenkin polyspherical coordinate system [6, Appendix B and
references therein], and that $\zeta=(r^{2}+r^{\prime 2})/(2rr^{\prime})$ is
defined in (3.11). We now give the Gegenbauer expansion for a logarithmic
fundamental solution of the polyharmonic equation in Vilenkin polyspherical
coordinate systems.

######  Theorem 5.2.

Let ${\bf x},{{\bf x}^{\prime}}\in{\mathbb{R}}^{d}$ with $d\in 2{\mathbb{N}}$,
$p=k-d/2\in{\mathbb{N}}_{0}$. Then the Gegenbauer expansion of a logarithmic
fundamental solution of the polyharmonic equation ${\mathfrak{l}}_{k}^{d}$ on
even-dimensional Euclidean space ${\mathbb{R}}^{d}$ in a Vilenkin
polyspherical coordinate system is given by

$\displaystyle{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})=i\frac{2^{\mu-\frac{1}{2}}}{\sqrt{\pi}}p!\Gamma(\mu)(\zeta^{2}-1)^{\frac{p+\mu}{2}+\frac{1}{4}}\left(2rr^{\prime}\right)^{p}\Biggl{[}\left(\log\left(rr^{\prime}\right)+\log{\left(\zeta+\sqrt{\zeta^{2}-1}\right)}+2H_{2p+2\mu}+H_{p}-H_{p+\mu}-2\beta_{p,d}\right)$
$\displaystyle\hskip
34.14322pt\times(-1)^{p+\mu+1}\sum_{l=0}^{p}\frac{(-1)^{l}(l+\mu)C_{l}^{\mu}(\cos\gamma)}{(p-l)!(l+p+2\mu)!}Q_{l+\mu-\frac{1}{2}}^{p+\mu+\frac{1}{2}}(\zeta).$
$\displaystyle\hskip
5.69046pt+(-1)^{p+\mu}\sum_{l=0}^{p}\frac{(-1)^{l}(l+\mu)C_{l}^{\mu}(\cos\gamma)}{(p-l)!(p+l+2\mu)!}\left(H_{p+l+2\mu}+H_{p-l}\right)Q_{l+\mu-\frac{1}{2}}^{p+\mu+\frac{1}{2}}(\zeta)$
$\displaystyle\hskip
5.69046pt+(-1)^{p+\mu+1}\sum_{l=0}^{p-1}\frac{(-1)^{l}(l+\mu)C_{l}^{\mu}(\cos\gamma)}{(p+l+2\mu)!}$
$\displaystyle\hskip
34.14322pt\times\sum_{k=0}^{p-l-1}\frac{(2l+2k+2\mu+1)}{k!(p-l-k)(p+l+k+2\mu+1)}\left[1+\frac{k!(p+l+2\mu)!}{(k+2l+2\mu)!(p-l)!}\right]Q_{l+\mu-\frac{1}{2}}^{k+l+\mu+\frac{1}{2}}(\zeta)$
$\displaystyle\hskip
5.69046pt+(-1)^{p+\mu+1}\sum_{l=0}^{p}\frac{(-1)^{l}(l+\mu)C_{l}^{\mu}(\cos\gamma)}{(p-l)!}\sum_{k=0}^{l+\mu-1}\frac{2k+1}{(l+k+\mu)!(p+\mu-k)(p+k+\mu+1)}Q_{l+\mu-\frac{1}{2}}^{k+\frac{1}{2}}(\zeta)$
$\displaystyle\hskip
5.69046pt+(-1)^{\mu}\sum_{l=p+1}^{\infty}\frac{(l+\mu)(l-p-1)!C_{l}^{\mu}(\cos\gamma)}{(p+l+2\mu)!}Q_{l+\mu-\frac{1}{2}}^{p+\mu+\frac{1}{2}}(\zeta)\Biggr{]}.$
(5.2)

###### Proof.

Applying the identities (4.6) and (4.4) to (3.16) completes the proof. ∎

For the polyharmonic equation on even-dimensional Euclidean space
${\mathbb{R}}^{d}$ with $k\geq\frac{d}{2}$, apart from multiplicative
constants, the algebraic expression for a binomial fundamental solution of the
polyharmonic equation
${\mathfrak{j}}_{k}^{d}:{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$
is given by ${\mathfrak{j}}_{k}^{d}({\bf x},{{\bf x}^{\prime}}):=\|{\bf
x}-{{\bf x}^{\prime}}\|^{2k-d},$ with $2k-d\in 2{\mathbb{N}}$. By expressing
${\mathfrak{j}}_{k}^{d}$ in a rotationally-invariant coordinate system one has
(3.18) which leads to the following result for an azimuthal Fourier expansion
of the binomial fundamental solution of the polyharmonic equation.

######  Theorem 5.3.

Let ${\bf x},{{\bf x}^{\prime}}\in{\mathbb{R}}^{d}$ with $d\in 2{\mathbb{N}}$,
$p=k-d/2\in{\mathbb{N}}_{0}$. Then the azimuthal Fourier expansion of a
binomial fundamental solution of the polyharmonic equation
${\mathfrak{j}}_{k}^{d}$ on Euclidean space ${\mathbb{R}}^{d}$ is given by

$\displaystyle{\mathfrak{j}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})=i(-1)^{p+1}\sqrt{\frac{2}{\pi}}p!(2RR^{\prime})^{p}{(\chi^{2}-1)^{\frac{p}{2}+\frac{1}{4}}}\sum_{n=0}^{p}\frac{(-1)^{n}\epsilon_{n}\cos\left(n(\phi-\phi^{\prime})\right)}{(p-n)!(p+n)!}Q_{n-\frac{1}{2}}^{p+\frac{1}{2}}(\chi).$
(5.3)

###### Proof.

Starting with (3.18), and using the binomial expansion (4.2) with the Fourier
cosine representation of the Chebyshev polynomials of the first kind completes
the proof. ∎

The expression ${\mathfrak{j}}_{k}^{d}$ can also be represented in any
Vilenkin polyspherical coordinate system (3.19) which likewise can be expanded
using Gegenbauer polynomials. This is presented as in the following result.

######  Theorem 5.4.

Let ${\bf x},{{\bf x}^{\prime}}\in{\mathbb{R}}^{d}$ with $d\in 2{\mathbb{N}}$,
$p=k-d/2\in{\mathbb{N}}_{0}$. Then the Gegenbauer expansion of a binomial
fundamental solution of the polyharmonic equation ${\mathfrak{j}}_{k}^{d}$ on
Euclidean space ${\mathbb{R}}^{d}$ is given by

$\displaystyle{\mathfrak{j}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})=i(-1)^{p-\frac{d}{2}}\frac{2^{\frac{d-1}{2}}}{\sqrt{\pi}}p!(\tfrac{d}{2}-2)!(2rr^{\prime})^{p}\left(\frac{r_{>}^{2}-r_{<}^{2}}{2rr^{\prime}}\right)^{p+\frac{d-1}{2}}$
$\displaystyle\hskip
170.71652pt\times\sum_{n=0}^{p}\frac{(-1)^{n}(n+\frac{d}{2}-1)C_{n}^{\frac{d}{2}-1}(\cos\gamma)}{(p-n)!(n+p+d-2)!}Q_{n+\frac{d-3}{2}}^{p+\frac{d-1}{2}}(\zeta).$
(5.4)

###### Proof.

The Gegenbauer polynomial expansion of Lemma 4.4 with $\mu=d/2-1$, is combined
with a fundamental solution written in a Vilenkin polyspherical coordinate
system and using (3.13) completes the proof. ∎

##  6 Global analysis on ${\mathbb{R}}^{d}$ of standard and generalized Hopf
Vilenkin polyspherical coordinate systems

Now we study some of the particular details which will arise in the study of
binomial and logarithmic fundamental solutions of the polyharmonic equation in
standard polyspherical coordinates and generalized Hopf coordinates. We will
utilize the addition theorem for hyperspherical harmonics to expand the
critical Gegenbauer polynomial (that with an order equal to $d/2-1$) over the
product of normalized hyperspherical harmonics in that particular Vilenkin
polyspherical coordinate system. The reason we used the critical order $d/2-1$
is that these Gegenbauer polynomials with this particular order provide a
basis for analytic solutions on ${\mathbf{S}}_{r}^{d-1}$. For $d\geq 3$, the
addition theorem for hyperspherical harmonics is given by (for a proof see
[26], [10, §10.2.1])

$C_{n}^{d/2-1}(\cos\gamma)=\frac{2(d-2)\pi^{d/2}}{(2n+d-2)\Gamma(d/2)}\sum_{K}Y_{n}^{K}(\widehat{\bf
x})\overline{Y_{n}^{K}({\widehat{\bf x}^{\prime}})},$ (6.1)

where $K$ stands for a set of $(d-2)$-quantum numbers identifying harmonics
for a given value of ${n}\in{\mathbb{N}}_{0}$, and $\cos\gamma$ is the cosine
of the separation angle (3.12) between two arbitrary vectors ${\bf x},{{\bf
x}^{\prime}}\in{\mathbb{R}}^{d}$. The functions
$Y_{{n}}^{K}:{\mathbf{S}}^{d-1}\to{\mathbb{C}}$ are the normalized
hyperspherical harmonics. Normalization of the hyperspherical harmonics is
achieved through the integral

$\int_{{\mathbf{S}}^{d-1}}Y_{n}^{K}(\widehat{\bf
x})\overline{Y_{n}^{K}(\widehat{\bf x})}d\Omega=1,$

where $d\Omega$ is the Riemannian volume measure on ${\mathbf{S}}^{d-1}$.

First we will treat standard polyspherical coordinates and then we will treat
generalized Hopf coordinates. Both of these polyspherical coordinate systems
and many of their various properties are described in [6, Appendix B].

###  6.1 Standard polyspherical coordinates

Here we review details connected with standard Vilenkin polyspherical
coordinates. These coordinates and as well the general Vilenkin polyspherical
coordinates are described carefully in [6, Appendix B], and we will not depart
from the description and usage of standard polyspherical coordinates described
therein. Standard polyspherical coordinates are a generalization of the
spherical coordinate system that is most commonly encountered in multi-
dimensional calculus. What we refer to as standard polyspherical coordinates
are given by

$\displaystyle x_{1}=r\cos\theta_{1},$ $\displaystyle
x_{2}=r\sin\theta_{1}\cos\theta_{2},$ $\displaystyle
x_{3}=r\sin\theta_{1}\sin\theta_{2}\cos\theta_{3},$
$\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$ $\displaystyle
x_{d-2}=r\sin\theta_{1}\cdots\sin\theta_{d-3}\cos\theta_{d-2},$ $\displaystyle
x_{d-1}=r\sin\theta_{1}\cdots\sin\theta_{d-3}\sin\theta_{d-2}\cos\phi,$
$\displaystyle
x_{d}=r\sin\theta_{1}\cdots\sin\theta_{d-3}\sin\theta_{d-2}\sin\phi,$ (6.2)

where $\theta_{i}\in[0,\pi]$ for $1\leq i\leq d-2$ and $\phi\in[-\pi,\pi)$. In
standard polyspherical coordinates, the normalized hyperspherical harmonics
can be written as [6, (B.19)]

$Y_{l}^{K}(\hat{{\bf
x}})=\frac{{\mathrm{e}}^{im\phi}}{\sqrt{2\pi}}\prod_{j=1}^{d-2}\Theta_{j}^{d}(l_{j},l_{j+1};\theta_{j}),$
(6.3)

where [6, (B.20)]

$\displaystyle\Theta_{j}^{d}(l_{j},l_{j+1};\theta_{j}):=$
$\displaystyle\frac{\Gamma\left(l_{j+1}+\frac{d-j+1}{2}\right)}{2l_{j+1}+d-j-1}\sqrt{\frac{2^{2l_{j+1}+d-j-1}(2l_{j}+d-j-1)(l_{j}-l_{j+1})!}{\pi(l_{j}+l_{j+1}+d-j-2)!}}$
$\displaystyle\times(\sin\theta_{j})^{l_{j+1}}C_{l_{j}-l_{j+1}}^{l_{j+1}+(d-j-1)/2}(\cos\theta_{j}).$
(6.4)

The addition theorem for hyperspherical harmonics (6.1) involves the product
$Y_{l}^{K}({\bf x})\overline{Y_{l}^{K}({{\bf x}^{\prime}})}$, so we introduce
a convenient notation, namely
$\Omega_{k}^{d}:{\mathbb{N}}_{0}\times{\mathbb{Z}}\times[0,\pi]^{2}\to{\mathbb{R}}$
is defined by

$\Omega_{k}^{d}\left(l_{k},l_{k+1};\\!\begin{array}[]{c}{\theta_{k}}\\\\[3.0pt]
{\theta_{k}^{\prime}}\end{array}\right):=\Theta_{k}^{d}(l_{k},l_{k+1};\theta_{k})\Theta_{k}^{d}(l_{k},l_{k+1};\theta_{k}^{\prime}).$
(6.5)

####  6.1.1 Standard polyspherical coordinates multi-sum reversal lemmas

An important ingredient in the production of the binomial and logarithmic
addition functions is reversing the order of the multi-sums that appear when
the Gegenbauer polynomials are expanded as multi-sums of hyperspherical
harmonics. This allows us to compare the Fourier coefficients relying on the
azimuthal quantum number $m$.

######  Lemma 6.1.

Let $p\in\mathbb{N}_{0}$. Then the multi-sum $\sf Y_{1}$, over the allowed
quantum numbers for standard polyspherical hyperspherical harmonics (6.3)
defined by

$\displaystyle{\sf Y}_{1}:=\sum_{l=0}^{p}\sum_{K}$
$\displaystyle=\sum_{l=0}^{p}\sum_{l_{2}=0}^{l}\cdots\sum_{l_{d-2}=0}^{l_{d-3}}\sum_{m=0}^{l_{d-2}},$
(6.6)

can then be re-expressed with the sum order reversed as

$\displaystyle{\sf
Y}_{1}=\sum_{m=0}^{p}\sum_{l_{d-2}=m}^{p}\cdots\sum_{l_{2}=l_{3}}^{p}\sum_{l=l_{2}}^{p}.$
(6.7)

###### Proof.

To reverse the multi-sum, the upper and lower bounds of the indices need to be
determined. Each index of the original multi-sum can have values between 0 and
$p$ inclusively. However, the upper bounds of the original multi-sum gives the
following constraint:

$\displaystyle 0\leq m\leq l_{d-2}\leq...\leq l_{2}\leq l.$ (6.8)

When the indices are reversed, this constraint determines the new lower bound
of each of the indices. Since there are no other constraints, the upper bound
for each index will be $p$. ∎

######  Lemma 6.2.

Let $p\in\mathbb{N}_{0}$. Then the multi-sum ${\sf Y}_{2}$ over quantum
numbers for standard polyspherical harmonics defined by

$\displaystyle{\sf Y}_{2}$
$\displaystyle:=\sum_{l=p+1}^{\infty}\sum_{K}=\sum_{l=p+1}^{\infty}\sum_{l_{2}=0}^{l}\cdots\sum_{l_{d-2}=0}^{l_{d-3}}\sum_{m=0}^{l_{2}}.$

can then be re-expressed with the sum order reversed as

$\displaystyle{\sf
Y}_{2}=\sum_{m=0}^{p}\sum_{l_{d-2}=m}^{\infty}\cdots\sum_{l_{2}=l_{3}}^{\infty}\sum_{l=\max(l_{2},p+1)}^{\infty}+\sum_{m=p+1}^{\infty}\sum_{l_{d-2}=m}^{\infty}\cdots\sum_{l_{2}=l_{3}}^{\infty}\sum_{l=l_{2}}^{\infty}.$
(6.9)

###### Proof.

First it is noted that each of the indices in the original multi-sum, except
for $l$, will have a lower bound of 0 and an upper bound of infinity. The
index $l$ will have the same upper bound, but a lower bound of $p+1$. Like the
previous lemma, the upper bounds of the original multi-sum will give us the
constraint, (6.8), which determines the lower bounds of the reversed multi-
sum. The combination of the following two lower bounds:

$\displaystyle l_{2}\leq l,\quad p+1\leq l,$

yields the lower bound $\max(p+1,l_{2})$. With this, the multi-sum can be
reversed:

$\displaystyle{\sf
Y_{2}}=\sum_{m=0}^{\infty}\sum_{l_{d-2}=m}^{\infty}\cdots\sum_{l_{2}=l_{3}}^{\infty}\sum_{l=\text{max}(l_{2},p+1)}^{\infty}.$

In later theorems, the scenarios of $m\leq p$ and $m\geq p+1$ need to be
treated separately. With the index $m$ being the first sum of the multi-sum,
this split is uncomplicated. When $m\geq p+1$, we can drop the maximum
function on the lower bound of $l$. ∎

###  6.2 Generalized Hopf coordinates

Generalized Hopf coordinates are a type of Vilenkin polyspherical coordinates
on ${\mathbb{R}}^{d}$ with $d=2^{q}$ for $q\geq 1$. They are Vilenkin
polyspherical orthogonal curvilinear coordinates with one radial coordinate
$r\in[0,\infty)$, and $(d-1)$-angular coordinates which together parametrize
points on ${\mathbf{S}}_{r}^{d-1}$ the $(d-1)$-dimensional real hypersphere
with radius $r$. Of the $(d-1)$-angular coordinates
$(d/2\\!-\\!1)$-$\vartheta$ coordinates take values in $[0,\frac{1}{2}\pi]$,
and the other $(d/2)$-$\phi$ coordinates are of azimuthal type and take values
in $[-\pi,\pi)$. For a careful treatment of generalized Hopf coordinates, see
[6, Appendix B].

In this paper we depart slightly from our previous description of generalized
Hopf coordinates—we have adopted a reversed azimuthal identification for the
azimuthal angles and their corresponding quantum numbers. In particular if one
considers the collection of angles in generalized Hopf coordinates given by
some vector of angles
${\bf\boldsymbol{\Theta}}:=(\Theta_{1},\ldots,\Theta_{d-1})$ with vector of
corresponding quantum numbers ${\bf p}:=(p_{1},\ldots,p_{d-1})$. In our
previous paper, these were ordered as

$\displaystyle\boldsymbol{\Theta}=(\vartheta_{1},\ldots,\vartheta_{d/2-1},\phi_{1},\ldots,\phi_{d/2}),\quad{\bf
p}=(l_{1},\ldots,l_{d/2-1},m_{1},\ldots,m_{d/2}),$ (6.10)

and in the current paper we order them as

$\displaystyle\boldsymbol{\Theta}=(\vartheta_{1},\ldots,\vartheta_{d/2-1},\phi_{d/2},\ldots,\phi_{1}),\quad{\bf
p}=(l_{1},\ldots,l_{d/2-1},m_{d/2},\ldots,m_{1})$ (6.11)

(see Figure 1 as compared to [6, Figure 5]).

These coordinates generalize two-dimensional polar coordinates (see Figure 1a)

$\displaystyle x_{1}=r\cos\phi,\qquad x_{2}=r\sin\phi,$ (6.12)

and four-dimensional Hopf coordinates (see Figure 1b)

$\displaystyle x_{1}=r\cos\vartheta\cos\phi_{2},\qquad$ $\displaystyle
x_{2}=r\cos\vartheta\sin\phi_{2},$ $\displaystyle
x_{3}=r\sin\vartheta\cos\phi_{1},\qquad$ $\displaystyle
x_{4}=r\sin\vartheta\sin\phi_{1}.$ (6.13)

See Figure 1c for a Vilenkin tree of the $d=8$ generalized Hopf coordinates.
In general the transformation formulae to Cartesian coordinates for
generalized Hopf coordinates is given by

$\displaystyle
x_{1}=r\cos\vartheta_{1}\cos\vartheta_{2}\cos\vartheta_{4}\cos\vartheta_{8}\cdots\cos\vartheta_{2^{q-2}}\cos\phi_{2^{q-1}},$
$\displaystyle
x_{2}=r\cos\vartheta_{1}\cos\vartheta_{2}\cos\vartheta_{4}\cos\vartheta_{8}\cdots\cos\vartheta_{2^{q-2}}\sin\phi_{2^{q-1}},$
$\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$
$\displaystyle
x_{2^{q-1}-1}=r\cos\vartheta_{1}\sin\vartheta_{2}\sin\vartheta_{5}\sin\vartheta_{11}\cdots\sin\vartheta_{3\cdot
2^{q-3}-1}\cos\phi_{2^{q-2}+1},$ $\displaystyle
x_{2^{q-1}}=r\cos\vartheta_{1}\sin\vartheta_{2}\sin\vartheta_{5}\sin\vartheta_{11}\cdots\sin\vartheta_{3\cdot
2^{q-3}-1}\sin\phi_{2^{q-2}+1},$ $\displaystyle
x_{2^{q-1}+1}=r\sin\vartheta_{1}\cos\vartheta_{3}\cos\vartheta_{6}\cos\vartheta_{12}\cdots\cos\vartheta_{3\cdot
2^{q-3}}\cos\phi_{2^{q-2}},$ $\displaystyle
x_{2^{q-1}+2}=r\sin\vartheta_{1}\cos\vartheta_{3}\cos\vartheta_{6}\cos\vartheta_{12}\cdots\cos\vartheta_{3\cdot
2^{q-3}}\sin\phi_{2^{q-2}},$
$\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$
$\displaystyle
x_{2^{q}-1}=r\sin\vartheta_{1}\sin\vartheta_{3}\sin\vartheta_{7}\sin\vartheta_{17}\cdots\sin\vartheta_{2^{q-1}-1}\cos\phi_{1},$
$\displaystyle
x_{2^{q}}=r\sin\vartheta_{1}\sin\vartheta_{3}\sin\vartheta_{7}\sin\vartheta_{17}\cdots\sin\vartheta_{2^{q-1}-1}\sin\phi_{1},$
(6.14)

where $\vartheta_{j}\in\big{[}0,\frac{1}{2}\pi\big{]}$ for $1\leq j\leq
2^{q-1}\\!-\\!1$ and $\phi_{k}\in[-\pi,\pi)$ for $1\leq k\leq 2^{q-1}$.
Generalized Hopf coordinates are unique in that they correspond to the only
trees which contain only themselves in their equivalence class (see [6,
(B.2)]). These coordinate systems have separated harmonic eigenfunctions which
are given in terms of complex exponentials of the azimuthal angles, and for
$q\geq 2$, non-symmetric Jacobi polynomials for the quantum numbers
corresponding to the $\vartheta$-angles.

Figure 1: This figure is a tree diagram for polyspherical generalized Hopf
coordinates of on ${\mathbb{R}}^{2^{q}}$ with $q=1,2,3,4$ for (a), (b), (c),
(d) respectively. The first $(2^{q-1}\\!-\\!1)$-branching nodes correspond to
the angles $\vartheta_{j}\in\big{[}0,\frac{1}{2}\pi\big{]}$ and quantum
numbers $l_{j}\in{\mathbb{N}}_{0}$, $1\leq j\leq d/2\\!-\\!1$. The following
$(2^{q-1})$-branching nodes correspond to the angles $\phi_{k}\in[-\pi,\pi)$
and quantum numbers $m_{k}\in{\mathbb{Z}}$, $1\leq k\leq d/2$. These
coordinates correspond to transformation (6.14).

The cosine of the separation angle (3.12) in these coordinates may be given as
follows. Define the symbol ${}_{q}{\sf G}_{s}^{r}\in[-1,1]$, where $0\leq
s\leq q$ and $1\leq r\leq 2^{q}-1$, by the recursive formula

${}_{q}{\sf
G}_{s}^{r}=\cos\vartheta_{r-1+2^{q-s}}\cos\vartheta_{r-1+2^{q-s}}^{\prime}\
{}_{q}{\sf
G}_{s-1}^{2r-1}+\sin\vartheta_{r-1+2^{q-s}}\sin\vartheta_{r-1+2^{q-s}}^{\prime}\
{}_{q}{\sf G}_{s-1}^{2r},$

with ${}_{q}{\sf G}_{0}^{i}=1$. Then the cosine of the separation angle is
given by

$\cos\gamma={}_{q}{\sf G}_{q}^{1}.$

Note that through the identification $\phi_{k}=\Theta_{d/2+1-k}$, where $1\leq
k\leq 2^{q-1}$, then ${}_{q}{\sf G}_{1}^{i}=\cos(\phi_{i}-\phi_{i}^{\prime})$.
Thus, this shows one may stop this recursion at $s=1$.

In regard to the quantum numbers in generalized Hopf coordinates, denote the
meridional quantum numbers $l_{k}\in{\mathbb{N}}_{0}$ and the azimuthal
quantum numbers $m_{j}\in{\mathbb{Z}}$, such that $1\leq k\leq d/2-1$, $1\leq
j\leq d/2$. Define ${\mathcal{L}}:=(l_{1},l_{2},\ldots,l_{{d/2}-1})$,
${\mathcal{M}}:=(m_{d/2},\ldots,m_{1})$ where $l:=l_{1}$ and $m:=m_{1}$. In
these coordinates, it is convenient to relate the meridional quantum numbers
$l_{k}$ to corresponding surrogate quantum numbers $n_{k}$ using
$l_{k}=2n_{k}+l_{2k}+l_{2k+1}$, and we define
${\mathcal{N}}:=(n_{1},n_{2},\ldots,n_{{d/2}-1})$, where $n:=n_{1}$, and
$n_{k}\in{\mathbb{N}}_{0}$ for all $1\leq k\leq d/2-1$. For convenience define
$N:=\sum_{k=1}^{d/2-1}n_{k}$, $M:=\sum_{j=1}^{d/2}m_{j}$. Note that one can
always write

$\displaystyle l_{1}=2N+M,$ $\displaystyle
l_{2}=2\sum_{j=2}^{\log_{2}(d/2)}\sum_{k=0}^{2^{j-2}-1}n_{2^{j-1}+k}+\sum_{j={d/4}+1}^{d/2}m_{j},$
$\displaystyle
l_{3}=2\sum_{j=2}^{\log_{2}(d/2)}\sum_{k=0}^{2^{j-2}-1}n_{2^{j-2}+2^{j-1}+k}+\sum_{j=1}^{d/4}m_{j}.$

In generalized Hopf coordinates, the normalized hyperspherical harmonics are
given by [6, (B.21)]

$Y_{l}^{K}({\bf x})=\frac{\prod_{1\leq j\leq
d/2}\exp(im_{j}\phi_{j})}{\sqrt{2}\,\pi^{d/4}}\Upsilon_{1}^{\log_{2}d}\left(\begin{array}[]{c}{n}\\\
{l_{2},l_{3}}\end{array};{\vartheta}\right)\times\cdots\times\Upsilon_{{d/2}-1}^{\log_{2}d}\left(\begin{array}[]{c}{n_{{d/2}-1}}\\\
{m_{2},m}\end{array};{\vartheta_{{d/2}-1}}\right),$

where
$\Upsilon_{k}^{\log_{2}d}:{\mathbb{N}}_{0}^{3}\times[0,\frac{1}{2}\pi]\to{\mathbb{R}}$
is defined by

$\displaystyle\Upsilon_{k}^{\log_{2}d}\left(\begin{array}[]{c}{n_{k}}\\\
{l_{2k},\l_{2k+1}}\end{array};{\vartheta_{k}}\right):=\sqrt{\frac{(2n_{k}+\alpha+\beta+1)(n_{k}+\alpha+\beta)!n_{k}!}{(n_{k}+\alpha)!(n_{k}+\beta)!}}$
(6.17) $\displaystyle\hskip
142.26378pt\times(\cos\vartheta_{k})^{l_{2k}}(\sin\vartheta_{k})^{l_{2k+1}}P_{n}^{(\beta,\alpha)}(\cos(2\vartheta_{k})),$
(6.18)
$\displaystyle\alpha=\alpha_{k}^{d}(l_{2k}):=l_{2k}-1+2^{\log_{2}(d/4)-\lfloor\log_{2}k\rfloor},$
$\displaystyle\beta=\beta_{k}^{d}(l_{2k+1}):=l_{2k+1}-1+2^{\log_{2}(d/4)-\lfloor\log_{2}k\rfloor}.$

######  Remark 6.1.

Note that if $d/4\leq k\leq d/2-1$ then
$-1+2^{\log_{2}(d/4)-\lfloor\log_{2}k\rfloor}=0$, hence
$\alpha,\beta,l_{2k},l_{2k+1}\in{\mathcal{M}}$. Also if $1\leq k\leq d/4-1$,
then $l_{2k},l_{2k+1}\in{\mathcal{L}}$.

The addition theorem for hyperspherical harmonics (6.1) involves the product
$Y_{l}^{K}({\bf x})\overline{Y_{l}^{K}({{\bf x}^{\prime}})}$, so we introduce
a convenient notation, namely
$\Psi_{k}^{\log_{2}d}:{\mathbb{N}}_{0}^{3}\times[0,\frac{1}{2}\pi]^{2}\to{\mathbb{R}}$
is defined by

$\Psi_{k}^{\log_{2}d}\left(\begin{array}[]{c}{n_{k}}\\\
{l_{2k},\l_{2k+1}}\end{array};\begin{array}[]{c}{\vartheta_{k}}\\\\[5.0pt]
{\vartheta_{k}^{\prime}}\end{array}\right):=\Upsilon_{k}^{\log_{2}d}\left(\begin{array}[]{c}{n_{k}}\\\
{l_{2k},\l_{2k+1}}\end{array};{\vartheta_{k}}\right)\Upsilon_{k}^{\log_{2}d}\left(\begin{array}[]{c}{n_{k}}\\\
{l_{2k},\l_{2k+1}}\end{array};{\vartheta_{k}^{\prime}}\right).$ (6.19)

######  Theorem 0.

In generalized Hopf coordinates, the product of normalized harmonics which
appears in the addition theorem for hyperspherical harmonics (6.1) is given by

$\displaystyle Y_{l}^{K}({\bf x})\overline{Y_{l}^{K}({{\bf
x}^{\prime}})}=\frac{1}{2\pi^{\frac{d}{2}}}\epsilon_{m}\cos(m(\phi-\phi^{\prime}))\epsilon_{m_{2}}\cos(m_{2}(\phi_{m_{2}}-\phi_{m_{2}}^{\prime}))\cdots\epsilon_{m_{d/2}}\cos(m_{d/2}(\phi_{d/2}-\phi_{d/2}^{\prime}))$
$\displaystyle\hskip
85.35826pt\times\Psi_{{d/2}-1}^{\log_{2}d}\left(\begin{array}[]{c}{n_{{d/2}-1}}\\\
{m_{2},m}\end{array};\begin{array}[]{c}{\vartheta_{{d/2}-1}}\\\\[5.0pt]
{\vartheta_{{d/2}-1}^{\prime}}\end{array}\right)\times\cdots\times\Psi_{1}^{\log_{2}d}\left(\begin{array}[]{c}{n}\\\
{l_{2},l_{3}}\end{array};\begin{array}[]{c}{\vartheta}\\\
{\vartheta^{\prime}}\end{array}\right).$ (6.28)

######  Proof 6.2.

In the sequel, we sum over all quantum numbers. If the coefficients of a
product of Fourier series over $m_{j}\in{\mathcal{M}}$, whose coefficients are
$f_{m_{j}}$, are invariant under sign reversal transformation, namely
$f_{-m_{j}}=f_{m_{j}}$, we can rewrite the product of complex exponentials as
a product of trigonometric cosine functions. This is accomplished using

$\sum_{m_{j}\in{\mathbb{Z}}}f_{j}\exp(im_{j}\psi_{j})=\sum_{m_{j}\in{\mathbb{N}}_{0}}\epsilon_{m}f_{j}\cos(m_{j}\psi_{j}),$

where $\epsilon_{m}=2-\delta_{m,0}$. The eigenfunctions $\Psi_{k}^{\log_{2}d}$
are invariant under this transformation, which can be verified by applying
(2.30), (2.31) for $d/4\leq k\leq d/2-1$ (see Remark 6.1).

####  6.2.1 Generalized Hopf multi-sum reversal lemmas

In generalized Hopf coordinates (as well as in the large-part of Vilenkin
polyspherical coordinate systems), the procedure for summing over the
degenerate quantum numbers can become somewhat technical. In the specialized
case of generalized Hopf coordinates we outline the procedure for describing
the multi-sums over the degenerate quantum numbers and derive descriptions of
those same multi-sums, but instead with the sum orders reversed.

######  Lemma 1.

Let $p\in{\mathbb{N}}_{0}$, $1\leq j\leq d/2$, $1\leq k\leq d/2-1$. Consider
the multi-sum ${\sf Z}_{1}$ defined by

$\displaystyle{\sf
Z}_{1}:=\sum_{l=0}^{p}\sum_{l_{2}=0}^{\infty}\cdots\sum_{l_{2^{q-1}-1}=0}^{\infty}\
\sum_{m_{2^{q-1}}=0}^{\infty}\cdots\sum_{m_{2}=0}^{\infty}\ \sum_{m=0}^{p}$
(6.29)

over ${\mathcal{L}}\cup{\mathcal{M}}$ with $0\leq m_{j}\leq p$, $0\leq
l_{k}\leq p$, where the sum over the $l_{k}\in{\mathcal{L}}$ quantum numbers
are restricted such that

$n_{k}=\frac{l_{k}-l_{2k}-l_{2k+1}}{2}\in{\mathbb{N}}_{0}.$

Then the multi-sum ${\sf Z}_{1}$ can be re-expressed as a multi-sum over
${\mathcal{N}}\cup{\mathcal{M}},$ with sum order reversed to obtain

$\displaystyle{\sf
Z}_{1}=\sum_{m=0}^{p}\cdots\sum_{m_{k}=0}^{p-{\scriptstyle\sum_{i=1}^{k-1}m_{i}}}\cdots\sum_{m_{2^{q-1}}=0}^{p-{\scriptstyle\sum_{i=1}^{2^{q-1}-1}m_{i}}}\
\sum_{n_{2^{q-1}-1}=0}^{\left\lfloor\frac{p-{\scriptstyle\sum_{i=1}^{2^{q-1}}m_{i}}}{2}\right\rfloor}\cdots\sum_{n=0}^{\left\lfloor\frac{p-{\scriptstyle\sum_{i=1}^{2^{q-1}}m_{i}}-2\sum_{i=2}^{2^{q-1}-1}n_{i}}{2}\right\rfloor}.$
(6.30)

######  Proof 6.3.

Let $q$ denote an integer at least 2. Put $d=2^{q}$ and suppose also that $p$
is a positive integer. Let $\Lambda\subseteq{\mathbb{R}}^{d-1}$ consisting of
$(d-1)$-tuples $\bar{l}=(l_{1},\ldots,l_{d-1})$ satisfying the following
conditions: (i) For $k$ satisfying $1\leq k\leq d-1$, $0\leq l_{k}\leq p$;
(ii) For $k$ satisfying $1\leq k\leq\frac{d}{2}-1$, $l_{k}-l_{2k}-l_{2k+1}$ is
a nonnegative, even integer. Let $S\subseteq{\mathbb{R}}^{d-1}$ denote the
convex polytope defined by the following inequalities:

$\displaystyle l_{1}\leq p,$ $\displaystyle l_{k}\geq
l_{2k}+l_{2k+1}\qquad(1\leq k<\tfrac{d}{2}),$ $\displaystyle l_{k}\geq
0\qquad(\tfrac{d}{2}\leq k<d).$

If $L$ denotes the lattice $L\subseteq{\mathbb{Z}}^{d-1}$ consisting of
$(d-1)$-tuples of integers for which $l_{k}-l_{2k}-l_{2k+1}$ is even for
$1\leq k<\frac{d}{2}$, then $\Lambda=L\cap S$. (The polytope $S$ is a
$(d-1)$-simplex, as shown below.) For $1\leq k<\frac{d}{2}$, put
$n_{k}=\frac{1}{2}(l_{k}-l_{2k}-l_{2k+1})$; for $\frac{d}{2}\leq k<d$, put
$n_{k}=l_{k}$. The function
$\varphi:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}^{d}$ taking $\bar{l}$ to
$\bar{n}=(n_{1},\ldots,n_{d-1})$ is an invertible linear transformation. In
particular, $l_{1}$ is given in terms of the $n_{k}$’s by

$\displaystyle
l_{1}=2(n_{1}+\cdots+n_{\frac{d}{2}-1})+n_{\frac{d}{2}}+\cdots+n_{d-1}.$

The transformation takes $S$ to the polytope $T=\varphi(S)$ defined by

$\displaystyle n_{k}\geq 0\qquad(1\leq k<d),$ $\displaystyle
2(n_{1}+\cdots+n_{\frac{d}{2}-1})+n_{\frac{d}{2}}+\cdots+n_{d-1}\leq p.$

The polytope $T$ is a simplex, being the subset of the first orthant in
${\mathbb{R}}^{d-1}$ that is cut off by a hyperplane. Therefore the linearly
equivalent set $S$ is also a simplex. The function $\varphi$ takes $L$ to
${\mathbb{Z}}^{d-1}$ and consequently it takes $\Lambda:=L\cap S$ to
$\Gamma:={\mathbb{Z}}^{d-1}\cap T$. For a set $\\{x_{\bar{n}}\\}$ indexed by
$\Gamma$, one has
$\sum_{\bar{n}\in\Gamma}x_{\bar{n}}=\sum_{\bar{l}\in\Lambda}x_{\varphi(\bar{l})}$.
This completes the proof.

######  Lemma 2.

Let $p\in{\mathbb{N}}_{0}$. $1\leq j\leq{d/2}$, $1\leq k\leq{d/2}-1$. Consider
the multi-sum ${\sf Z}_{2}$ defined by

$\displaystyle{\sf
Z}_{2}:=\sum_{l=p+1}^{\infty}\sum_{l_{2}=0}^{\infty}\cdots\sum_{l_{2^{q-1}-1}=0}^{\infty}\
\sum_{m_{2^{q-1}}=0}^{\infty}\cdots\sum_{m_{2}=0}^{\infty}\ \sum_{m=0}^{p}$
(6.31)

over ${\mathcal{L}}\cup{\mathcal{M}}$ with $0\leq m_{j}\leq p$, $l_{k}\geq
p+1$, with the same restriction over the $l_{k}\in{\mathcal{L}}$ in the above
lemma. Then ${\sf Z}_{2}$ can be re-expressed as a multi-sum over
${\mathcal{N}}\cup{\mathcal{M}}$ with sum order reversed to obtain

$\displaystyle{\sf
Z}_{2}=\sum_{m=0}^{p}\sum_{m_{2}=0}^{\infty}\cdots\sum_{m_{2^{q-1}}=0}^{\infty}\
\sum_{n_{2^{q-1}-1}=0}^{\infty}\cdots\sum_{n_{2}=0}^{\infty}\
\sum_{n=\max\left(0,\left\lfloor\frac{p-{\scriptstyle\sum_{i=1}^{2^{q-1}}m_{i}}-2\sum_{i=2}^{2^{q-1}-1}n_{i}}{2}\right\rfloor+1\right)}^{\infty}.$
(6.32)

######  Proof 6.4.

Let $S^{\prime}\subseteq{\mathbb{R}}^{d-1}$ denote the convex polyhedron
defined by the following inequalities.

$\displaystyle l_{1}\geq p+1,$ $\displaystyle l_{k}\geq
l_{2k}+l_{2k+1}\qquad(1\leq k<\tfrac{d}{2}),$ $\displaystyle p\geq l_{k}\geq
0\qquad(\tfrac{d}{2}\leq k<d).$

The polyhedron $T^{\prime}=\varphi(S^{\prime})$ is given by the following
inequalities.

$\displaystyle n_{k}\geq 0\qquad(1\leq k<d),$ $\displaystyle n_{k}\leq
p\qquad(\frac{d}{2}\leq k<d),$ $\displaystyle
2(n_{1}+\cdots+n_{\frac{d}{2}-1})+n_{\frac{d}{2}}+\cdots+n_{d-1}\geq p+1.$

The set $\Lambda^{\prime}:=S^{\prime}\cap L$ is mapped to
$\Gamma^{\prime}:=T^{\prime}\cap{\mathbb{Z}}^{d-1}$ by $\varphi$. For a set
$\\{x_{\bar{n}}\\}$ indexed by $\Gamma^{\prime}$, one has
$\sum_{\bar{n}\in\Gamma^{\prime}}x_{\bar{n}}=\sum_{\bar{l}\in\Lambda^{\prime}}x_{\varphi(\bar{l})}$.
This completes the proof.

######  Lemma 3.

Let $m,p,k\in{\mathbb{N}}_{0}$ such that $p=k-d/2\in{\mathbb{N}}_{0}$, $d\in
2{\mathbb{N}}$ and $1\leq k\leq{d/2}-1$. Consider the following multi-sum over
${\mathcal{L}}\cup{\mathcal{M}}$ with $m\geq p+1$.

$\displaystyle{\sf
Z}_{3}:=\sum_{l=p+1}^{\infty}\sum_{l_{2}=0}^{\infty}\cdots\sum_{l_{{d/2}-1}=0}^{\infty}\sum_{m_{d/2}=0}^{\infty}\cdots\sum_{m_{2}=0}^{\infty}\sum_{m=p+1}^{\infty},$
(6.33)

with the same restriction over the $l_{k}\in{\mathcal{L}}$ in the above
lemmas. Then the multi-sum ${\sf Z}_{3}$ can be re-expressed as a multi-sum
over ${\mathcal{N}}\cup{\mathcal{M}}$ with sum order reversed to obtain

$\displaystyle{\sf
Z}_{3}=\sum_{m=p+1}^{\infty}\sum_{m_{2}=0}^{\infty}\cdots\sum_{m_{d/2}=0}^{\infty}\sum_{n_{{d/2}-1}=0}^{\infty}\cdots\sum_{n=0}^{\infty}.$
(6.34)

######  Proof 6.5.

Let $S^{\prime\prime}\subseteq{\mathbb{R}}^{d-1}$ denote the convex polyhedron
defined by the following inequalities:

$\displaystyle l_{k}\geq l_{2k}+l_{2k+1},\qquad(1\leq k<\tfrac{d}{2}),$
$\displaystyle l_{k}\geq 0,\qquad(\tfrac{d}{2}\leq k<d),$ $\displaystyle
l_{d-1}\geq p+1.$

The polyhedron $T^{\prime\prime}=\varphi(S^{\prime\prime})$ is given by the
following inequalities:

$\displaystyle n_{k}\geq 0\qquad(1\leq k<d),$ $\displaystyle n_{d-1}\geq p+1.$

The set $\Lambda^{\prime\prime}:=S^{\prime\prime}\cap L$ is mapped to
$\Gamma^{\prime\prime}:=T^{\prime\prime}\cap{\mathbb{Z}}^{d-1}$ by $\varphi$.
For a set $\\{x_{\bar{n}}\\}$ indexed by $\Gamma^{\prime\prime}$, one has

$\sum_{\bar{n}\in\Gamma^{\prime\prime}}x_{\bar{n}}=\sum_{\bar{l}\in\Lambda^{\prime\prime}}x_{\varphi(\bar{l})}.$

This completes the proof.

##  7  Binomial and logarithmic addition theorems for the azimuthal Fourier
coefficients

In even-dimensional Euclidean space ${\mathbb{R}}^{d}$, the kernels for a
fundamental solution of the polyharmonic equation are represented by a
function of the distance between two points (3.5), $\|{\bf x}-{{\bf
x}^{\prime}}\|$. In rotationally-invariant coordinates systems, one may
represent the distance between the source and observation points in terms of
the difference between azimuthal coordinates, namely (3.8). In Vilenkin’s
polyspherical coordinates, which are rotationally-invariant, one may also
represent this distance in terms of the separation angle, namely (3.10). In a
previous publication [6], addition theorems which arise when a fundamental
solution of the polyharmonic equation is of a power-law in Euclidean space
with odd-dimensions and for even-dimensions when $1\leq k<d/2$ were derived.
In this manuscript, we treat the even-dimensional case for $k\geq d/2$, in
which the functional dependence is either logarithmic or of binomial form. By
considering the equivalent azimuthal $\phi-\phi^{\prime}$ and separation angle
$\gamma$, Fourier and Gegenbauer expansions respectively of the kernels for
the polyharmonic equation in even-dimensional space, can derive addition
theorems for the azimuthal Fourier coefficients.

The procedure for developing the addition theorems in this section is as
follows. Let $p=k-d/2\in{\mathbb{N}}_{0},$ with
$f_{p}:{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$,
$g_{p}:({\mathbb{R}}^{d}\times{\mathbb{R}}^{d})\setminus\\{({\bf x},{\bf
x}):{\bf x}\in{\mathbb{R}}^{d}\\}\to{\mathbb{R}}$, defined such that
$f_{p}({\bf x},{{\bf x}^{\prime}}):=\|{\bf x}-{{\bf x}^{\prime}}\|^{2p}$,
$g_{p}({\bf x},{{\bf x}^{\prime}}):=\|{\bf x}-{{\bf
x}^{\prime}}\|^{2p}\log\|{\bf x}-{{\bf x}^{\prime}}\|$. First, express
$f_{p},g_{p}$ in terms of their azimuthal separation angle Fourier cosine
series using (4.2), (4.3). Call these the left-hand sides. Then express
$f_{p},g_{p}$ in terms of their separation angle Gegenbauer expansions given
using (4.4), (4.6) with $\mu=d/2-1$. Call these the right-hand sides. By using
the addition theorem for hyperspherical harmonics (6.1), we can expand the
right-hand sides in terms of a product of separable harmonics in a chosen
Vilenkin polyspherical coordinate system. Since Vilenkin’s polyspherical
coordinate systems are rotationally-invariant, one of the coordinates will
correspond to the chosen azimuthal separation angle which has been expanded
about on the left-hand side of the azimuthal Fourier expansion. To obtain the
addition theorem, one must re-arrange the multi-sum expression which arises on
the right-hand side so that the outermost sum is the sum over the relevant
azimuthal quantum number. Addition theorems are simply derived by comparing
the azimuthal Fourier coefficients on both sides.

In order to obtain binomial and logarithmic addition theorems in a Vilenkin
polyspherical coordinate system, we relate respectively $\|{\bf x}-{{\bf
x}^{\prime}}\|^{2p}$ and $\|{\bf x}-{{\bf x}^{\prime}}\|^{2p}\log\|{\bf
x}-{{\bf x}^{\prime}}|$, in terms of their Fourier cosine series over the
azimuthal separation angle, and their Gegenbauer polynomial expansions over
the separation angle (3.12). These equalities reduce respectively to

$\left(\chi-\cos(\phi-\phi^{\prime})\right)^{p}=\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}\left(\zeta-\cos\gamma\right)^{p},$
(7.1)

and

$\displaystyle\log(2RR^{\prime})\left(\chi-\cos(\phi-\phi^{\prime})\right)^{p}+(\chi-\cos(\phi-\phi^{\prime}))^{p}\log(\chi-\cos(\phi-\phi^{\prime}))$
$\displaystyle\hskip
56.9055pt=\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}\left(\log(2rr^{\prime})\left(\zeta-\cos\gamma\right)^{p}+(\zeta-\cos\gamma)^{p}\log(z-\cos\gamma)\right).$
(7.2)

###  7.1 Addition theorems in standard polyspherical coordinates

The binomial addition theorems in standard polyspherical coordinates are given
by the following two theorems.

######  Theorem 3.

Let $m\in{\mathbb{N}}_{0}$, $p=k-d/2\in{\mathbb{N}}_{0}$, $d\in
2{\mathbb{N}}$, $0\leq m\leq p$. Then

$\displaystyle Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)$
$\displaystyle=(-1)^{\frac{d}{2}-1+m}(2\pi)^{\frac{d}{2}-1}(\chi^{2}-1)^{-\frac{p}{2}-\frac{1}{4}}(p-m)!(p+m)!\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}\left(\frac{r_{>}^{2}-r_{<}^{2}}{2rr^{\prime}}\right)^{p+\frac{d-1}{2}}$
$\displaystyle\times\sum_{l_{d-2}=m}^{p}\Omega_{d-2}^{d}\left(l_{d-2},m;\\!\begin{array}[]{c}{\theta_{d-2}}\\\\[3.0pt]
{\theta_{d-2}^{\prime}}\end{array}\right)\cdots\sum_{l=l_{2}}^{p}\Omega_{1}^{d}\left(l,l_{2};\\!\begin{array}[]{c}{\theta}\\\\[0.0pt]
{\theta^{\prime}}\end{array}\right)\frac{(-1)^{l}}{(p-l)!(l+p+d-2)!}Q_{l+\frac{d-3}{2}}^{p+\frac{d-1}{2}}(\zeta)$

######  Proof 7.1.

This equality can be found by comparing the binomial expansions of Theorems
5.3 and 5.4. The Gegenbauer polynomials in Theorem 5.4 can be expanded with
the aid of (6.1). The normalized hyperspherical harmonics can be written in
standard polyspherical coordinates, as seen in (6.3). Using the concise
notation of (6.5), the Gegenbauer expansion of (6.1) can be written as
follows:

$C_{l}^{d/2-1}(\cos\gamma)=\frac{(d-2)\pi^{\frac{d}{2}-1}}{(2l+d-2)\Gamma(d/2)}\sum_{K}{\mathrm{e}}^{im(\phi-\phi^{\prime})}\Omega_{d-2}^{d}\left(l_{d-2},m;\\!\begin{array}[]{c}{\theta_{d-2}}\\\\[3.0pt]
{\theta_{d-2}^{\prime}}\end{array}\right)\cdots\Omega_{1}^{d}\left(l,l_{2};\\!\begin{array}[]{c}{\theta}\\\\[0.0pt]
{\theta^{\prime}}\end{array}\right).$ (7.3)

Now the expansion of Theorem 5.4 will contain a multi-sum. Our goal is to
compare the Fourier coefficients of the binomial expansions, this multi-sum
will need to be reversed. This reversal is shown in Lemma 6.1. This allows us
to compare the Fourier coefficients, and complete the proof.

######  Theorem 3.

Let $m\in{\mathbb{N}}_{0}$, $k\in{\mathbb{N}}$, $p=k-d/2\in{\mathbb{N}}_{0}$,
$d=4$ and $0\leq m\leq p$. Then

$\displaystyle Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)$
$\displaystyle=(-1)^{1-m}(\chi^{2}-1)^{-\frac{p}{2}-\frac{1}{4}}(p-m)!(p+m)!\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}\left(\frac{r_{>}^{2}-r_{<}^{2}}{2rr^{\prime}}\right)^{p+\frac{3}{2}}$
$\displaystyle\times\sum_{l_{2}=m}^{p}\frac{2^{2l_{2}}\left(l_{2}!\right)^{2}(2l_{2}+1)(l_{2}-m)!}{(l_{2}+m)!}\left(\sin{\theta}\sin{\theta^{\prime}}\right)^{l_{2}}{\sf
P}_{l_{2}}^{m}(\cos{\theta_{2}}){\sf
P}_{l_{2}}^{m}(\cos{\theta_{2}^{\prime}})$
$\displaystyle\times\sum_{l=l_{2}}^{p}\frac{(-1)^{l}(2l+2)(l-l_{2})!}{(p-l)!(l+p+2)!(l+l_{2}+1)!}C_{l-l_{2}}^{l_{2}+1}(\cos{\theta})C_{l-l_{2}}^{l_{2}+1}(\cos{\theta^{\prime}})Q_{l+\frac{1}{2}}^{p+\frac{3}{2}}(\zeta).$

######  Proof 7.2.

To prove this, take the result of Theorem 7.0 and use $d=4$. This simplifies
the multi-sum of $\Omega$ functions:

$\displaystyle\sum_{l_{d-2}}\Omega_{d-2}^{d}\left(l_{d-2},m;\\!\begin{array}[]{c}{\theta_{d-2}}\\\\[3.0pt]
{\theta_{d-2}^{\prime}}\end{array}\right)\cdots\sum_{l}\Omega_{1}^{d}\left(l,l_{2};\\!\begin{array}[]{c}{\theta}\\\\[0.0pt]
{\theta^{\prime}}\end{array}\right)\rightarrow\sum_{l_{2}}\Omega_{2}^{4}\left(l_{2},m;\\!\begin{array}[]{c}{\theta_{2}}\\\\[0.0pt]
{\theta^{\prime}_{2}}\end{array}\right)\sum_{l}\Omega_{1}^{4}\left(l,l_{2};\\!\begin{array}[]{c}{\theta}\\\\[0.0pt]
{\theta^{\prime}}\end{array}\right).$ (7.12)

By (6.5), we know the $\Omega$-functions are products of $\Theta$-functions,
which are defined in (6.1). A couple of the Gegenbauer polynomials that appear
can be rewritten as Ferrers functions using (2.27). This allows (7.12) to be
written as:

$\displaystyle\sum_{l_{2}}\Omega_{2}^{4}\left(l_{2},m;\\!\begin{array}[]{c}{\theta_{2}}\\\\[0.0pt]
{\theta^{\prime}_{2}}\end{array}\right)\sum_{l}\Omega_{1}^{4}\left(l,l_{2};\\!\begin{array}[]{c}{\theta}\\\\[0.0pt]
{\theta^{\prime}}\end{array}\right)=\frac{1}{\pi}\sum_{l_{2}}\frac{2^{2l_{2}}\left(l_{2}!\right)^{2}(2l_{2}+1)(l_{2}-m)!}{(l_{2}+m)!}$
(7.17) $\displaystyle\hskip
28.45274pt\times(\sin\theta\sin\theta^{\prime})^{l_{2}}{\sf
P}_{l_{2}}^{m}(\cos\theta_{2}){\sf
P}_{l_{2}}^{m}(\cos\theta^{\prime}_{2})\sum_{l}\frac{(l+1)(l-l_{2})!}{(l+l_{2}+1)!}C_{l-l_{2}}^{l_{2}+1}(\cos\theta)C_{l-l_{2}}^{l_{2}+1}(\cos\theta^{\prime}).$
(7.18)

Then simplification completes the proof.

In standard polyspherical coordinates, one has the following logarithmic
addition theorem for $0\leq m\leq p$.

######  Theorem 3.

Let $m\in{\mathbb{N}}_{0}$, $k\in{\mathbb{N}}$, $p=k-d/2\in{\mathbb{N}}_{0}$,
$d\in 2{\mathbb{N}}$ and $0\leq m\leq p$. Then

$\displaystyle\frac{1}{(p+m)!(p-m)!}\left(\log(RR^{\prime})+\log(\chi+\sqrt{\chi^{2}-1})+2H_{2p}-H_{p+m}-H_{p-m}\right)Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)$
$\displaystyle+\frac{1}{(p+m)!}\sum_{k=0}^{p-m-1}\frac{(2m+2k+1)}{k!(p-m-k)(p+m+k+1)}\left[1+\frac{k!(p+m)!}{(2m+k)!(p-m)!}\right]Q_{m-\frac{1}{2}}^{m+k+\frac{1}{2}}(\chi)$
$\displaystyle+\frac{1}{(p-m)!}\sum_{k=0}^{m-1}\frac{2k+1}{(m+k)!(p-k)(p+k+1)}Q_{m-\frac{1}{2}}^{k+\frac{1}{2}}(\chi)$
$\displaystyle=\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}(-1)^{m+{d/2}-1}(2\pi)^{{d/2}-1}\frac{(\zeta^{2}-1)^{\frac{p}{2}+\frac{d-1}{4}}}{(\chi^{2}-1)^{\frac{p}{2}+\frac{1}{4}}}$
$\displaystyle\times\biggl{\\{}\sum_{l_{d-2}=m}^{p}\Omega_{d-2}^{d}\left(l_{d-2},m;\\!\begin{array}[]{c}{\theta_{d-2}}\\\\[3.0pt]
{\theta_{d-2}^{\prime}}\end{array}\right)\cdots\sum_{l=l_{2}}^{p}\Omega_{1}^{d}\left(l,l_{2};\\!\begin{array}[]{c}{\theta}\\\\[0.0pt]
{\theta^{\prime}}\end{array}\right)$ (7.23) $\displaystyle\hskip
11.38092pt\times\biggl{[}\frac{(-1)^{l}Q_{l+\frac{d-3}{2}}^{p+\frac{d-1}{2}}(\zeta)}{(p-l)!(p+l+d-2)!}\left(\log(rr^{\prime})+\log(\zeta\\!+\\!\sqrt{\zeta^{2}-1})\\!+\\!2H_{2p+d-2}\\!+\\!H_{p}-H_{p+{d/2}-1}\\!-\\!H_{p+l+d-2}\\!-\\!H_{p-l}\right)$
$\displaystyle\hskip
25.6073pt+\frac{(-1)^{l}}{(p+l+d-2)!}\sum_{k=0}^{p-l-1}\frac{(2l+2k+d-1)}{k!(p-l-k)(p+l+k+d-1)}\left[1+\frac{k!(p+l+d-2)!}{(k+2l+d-2)!(p-l)!}\right]Q_{l+\frac{d-3}{2}}^{k+l+\frac{d-1}{2}}(\zeta)$
$\displaystyle\hskip
25.6073pt+\frac{(-1)^{l}}{(p-l)!}\sum_{k=0}^{l+{d/2}-2}\frac{2k+1}{(l+{d/2}+k-1)!(p+{d/2}-k-1)(p+{d/2}+k)}Q_{l+\frac{d-3}{2}}^{k+\frac{1}{2}}(\zeta)\biggr{]}$
$\displaystyle\hskip
11.38092pt+\sum_{l_{d-2}=m}^{\infty}\Omega_{d-2}^{d}\left(l_{d-2},m;\\!\begin{array}[]{c}{\theta_{d-2}}\\\\[3.0pt]
{\theta_{d-2}^{\prime}}\end{array}\right)\cdots\sum_{l_{2}=l_{3}}^{\infty}\Omega_{2}^{d}\left(l_{2},l_{3};\\!\begin{array}[]{c}{\theta_{2}}\\\\[3.0pt]
{\theta_{2}^{\prime}}\end{array}\right)$ (7.28) $\displaystyle\hskip
31.2982pt\times\sum_{l=\text{max}(p+1,l_{2})}^{\infty}\frac{(-1)^{p+1}(l-p-1)!}{(p+l+d-2)!}\Omega_{1}^{d}\left(l,l_{2};\\!\begin{array}[]{c}{\theta}\\\\[0.0pt]
{\theta^{\prime}}\end{array}\right)Q_{l+\frac{d-3}{2}}^{p+\frac{d-1}{2}}(\zeta)\biggr{\\}}.$
(7.31)

######  Proof 7.3.

To prove this theorem, we take the Fourier and Gegenbauer expansions of
logarithmic fundamental solutions, seen in Theorems 5.1 and 5.2 respectively.
For the Gegenbauer expansion, the substitution of $\mu=d/2-1$ is used. Again,
the Gegenbauer polynomials can be rewritten as in (7.3). The Gegenbauer
expansion contains three different multi-sums, one contains the sum of $l$
from zero to $p$, the next contains the sum of $l$ from zero to $p+1$, and the
last contains the sum of $l$ from $p+1$ to infinity. The first two of these
sums can be reversed as in Lemma (6.1), where the later $p$ can be replaced
with $p+1$. For the final multi-sum, Lemma 6.2 is used. Since only the region
$0\leq m\leq p$ is required, only the first term of the split multi-sum is
used. When comparing the Fourier coefficients of the expansions, only those
terms where $0\leq m\leq p$ are used. Thus only the first two parts of the
above multi-sum are used.

The following corollary results substituting $d=4$ in the above logarithmic
addition theorem (7.31).

######  Corollary 4.

Let $0\leq m\leq p$, $p=k-d/2\in{\mathbb{N}}_{0}$, $k\in{\mathbb{N}}$ and
$d=4$. Then

$\displaystyle\frac{1}{(p+m)!(p-m)!}\left(\log(RR^{\prime})+\log(\chi+\sqrt{\chi^{2}-1})+2H_{2p}-H_{p+m}-H_{p-m}\right)Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)$
$\displaystyle+\frac{1}{(p+m)!}\sum_{k=0}^{p-m-1}\frac{(2m+2k+1)}{k!(p-m-k)(p+m+k+1)}\left[1+\frac{k!(p+m)!}{(2m+k)!(p-m)!}\right]Q_{m-\frac{1}{2}}^{m+k+\frac{1}{2}}(\chi)$
$\displaystyle+\frac{1}{(p-m)!}\sum_{k=0}^{m-1}\frac{2k+1}{(m+k)!(p-k)(p+k+1)}Q_{m-\frac{1}{2}}^{k+\frac{1}{2}}(\chi)$
$\displaystyle=2\,(-1)^{m+1}\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}\frac{(\zeta^{2}-1)^{\frac{p}{2}+\frac{3}{4}}}{(\chi^{2}-1)^{\frac{p}{2}+\frac{1}{4}}}\biggl{\\{}\sum_{l_{2}=m}^{p}\frac{(2l_{2}+1)(l_{2}-m)!}{(l_{2}+m)!}{\sf
P}_{l_{2}}^{m}(\cos\theta_{2}){\sf
P}_{l_{2}}^{m}(\cos\theta_{2}^{\prime})2^{2l_{2}}(l_{2}!)^{2}(\sin\theta\sin\theta^{\prime})^{l_{2}}$
$\displaystyle\hskip
14.22636pt\times\sum_{l=l_{2}}^{p}\frac{(-1)^{l}(l+1)(l-l_{2})!}{(l+l_{2}+1)!}C_{l-l_{2}}^{l_{2}+1}(\cos\theta)C_{l-l_{2}}^{l_{2}+1}(\cos\theta^{\prime})$
$\displaystyle\hskip
28.45274pt\times\biggl{[}\frac{1}{(p-l)!(p+l+2)!}\left(\log(rr^{\prime})+\log(\zeta+\sqrt{\zeta^{2}-1})+2H_{2p+2}+H_{p}-H_{p+1}-H_{p+l+2}-H_{p-l}\right)Q_{l+{d/2}}^{p+\frac{3}{2}}(\zeta)$
$\displaystyle\hskip
42.67912pt+\frac{1}{(p+l+2)!}\sum_{k=0}^{p-l-1}\frac{(2l+2k+3)}{k!(p-l-k)(p+l+k+3)}\left[1+\frac{k!(p+l+2)!}{(k+2l+2)!(p-l)!}\right]Q_{l+\frac{1}{2}}^{k+l+\frac{3}{2}}(\zeta)$
$\displaystyle\hskip
42.67912pt+\frac{1}{(p-l)!}\sum_{k=0}^{l}\frac{2k+1}{(l+k+1)!(p-k+1)(p+k+2)}Q_{l+\frac{1}{2}}^{k+\frac{1}{2}}(\zeta)\biggr{]}$
$\displaystyle\hskip
14.22636pt+(-1)^{p+1}\sum_{l_{2}=m}^{\infty}\frac{(2l_{2}+1)(l_{2}-m)!}{(l_{2}+m)!}{\sf
P}_{l_{2}}^{m}(\cos\theta_{2}){\sf
P}_{l_{2}}^{m}(\cos\theta_{2}^{\prime})2^{2l_{2}}(l_{2}!)^{2}(\sin\theta\sin\theta^{\prime})^{l_{2}}$
$\displaystyle\hskip
28.45274pt\times\sum_{l=\max(l_{2},p+1)}^{\infty}\frac{(l+1)(l-l_{2})!}{(l+l_{2}+1)!}C_{l-l_{2}}^{l_{2}+1}(\cos\theta)C_{l-l_{2}}^{l_{2}+1}(\cos\theta^{\prime})\frac{(l-p-1)!}{(p+l+2)!}Q_{l+\frac{1}{2}}^{p+\frac{3}{2}}(\zeta)\biggr{\\}}.$
(7.32)

######  Proof 7.4.

To prove this, take the result of Theorem 7.2 for the $d=4$ case. The
$\Omega$-functions can be simplified as in the proof of Theorem 7.1, as seen
in (7.17). This completes the proof.

In standard polyspherical coordinates, one has the following logarithmic
addition theorem for $m\geq p+1$.

######  Theorem 4.

Let $m\in{\mathbb{N}}_{0}$, $p=k-d/2\in{\mathbb{N}}_{0}$, $d\in
2{\mathbb{N}}$, $m\geq p+1$. Then

$\displaystyle Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)$
$\displaystyle=(-1)^{\frac{d}{2}-1}(2\pi)^{\frac{d}{2}-1}(\zeta^{2}-1)^{\frac{p}{2}+\frac{d-1}{4}}(\chi^{2}-1)^{-\frac{p}{2}-\frac{1}{4}}\frac{(p+m)!}{(m-p-1)!}\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}$
$\displaystyle\times\sum_{l_{d-2}=p+1}^{\infty}\Omega_{d-2}^{d}\left(l_{d-2},m;\\!\begin{array}[]{c}{\theta_{d-2}}\\\\[3.0pt]
{\theta_{d-2}^{\prime}}\end{array}\right)\cdots\sum_{l=l_{2}}^{\infty}\Omega_{1}^{d}\left(l,l_{2};\\!\begin{array}[]{c}{\theta}\\\\[0.0pt]
{\theta^{\prime}}\end{array}\right)\frac{(l-p-1)!}{(p+l+d-2)!}Q_{l+\frac{d-3}{2}}^{p+\frac{d-1}{2}}(\zeta).$
(7.37)

######  Proof 7.5.

Again, this proofs starts with the equal logarithmic expansions of 5.1 and
5.2. With the Gegenbauer expansion, the polynomials are rewritten as seen in
(7.3). The Fourier coefficients of interest only appear in the multi-sum that
has $l$ summing over $p+1$ to infinity. This multi-sum can be seen in Lemma
6.2. The second term of this multi-sum contains values of $m$ that are greater
than $p$, which are the terms needed for the comparison to complete the proof.

######  Corollary 5.

Let $m\in{\mathbb{N}}_{0}$, $p=k-d/2\in{\mathbb{N}}_{0}$, $d=4$, $m\geq p+1$.
Then

$\displaystyle
Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)=-2(\zeta^{2}-1)^{\frac{p}{2}+\frac{3}{4}}(\chi^{2}-1)^{-\frac{p}{2}-\frac{1}{4}}\frac{(p+m)!}{(m-p-1)!}\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}$
$\displaystyle\hskip
14.22636pt\times\sum_{l_{2}=p+1}^{\infty}\frac{2^{2l_{2}}\left(l_{2}!\right)^{2}(2l_{2}+1)(l_{2}-m)!}{(l_{2}+m)!}$
$\displaystyle\hskip
21.33955pt\times(\sin\theta\sin\theta^{\prime})^{l_{2}}{\sf
P}_{l_{2}}^{m}(\cos\theta_{2}){\sf
P}_{l_{2}}^{m}(\cos\theta^{\prime}_{2})\sum_{l=l_{2}}^{\infty}\frac{(l+1)(l-l_{2})!}{(l+l_{2}+1)!}C_{l-l_{2}}^{l_{2}+1}(\cos\theta)C_{l-l_{2}}^{l_{2}+1}(\cos\theta^{\prime})Q_{l-\frac{1}{2}}^{p+\frac{3}{2}}(\zeta).$
(7.38)

######  Proof 7.6.

Start with (7.37) and set $d=4$. The $\Omega$-functions can be simplified as
in (7.17), which completes the proof.

###  7.2 Addition theorems in generalized Hopf coordinates

In generalized Hopf coordinates, one has the following binomial addition
theorem valid for $0\leq m\leq p$.

######  Theorem 5.

Let $p,m\in{\mathbb{N}}_{0}$, $0\leq m\leq p$, $d=2^{q}$, $q\geq 2$. Then

$\displaystyle
Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)=(-1)^{m+{d/2}-1}2^{{d/2}-1}(p-m)!(p+m)!(\chi^{2}-1)^{-\frac{p}{2}-\frac{1}{4}}\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}\left(\frac{r_{>}^{2}-r_{<}^{2}}{2rr^{\prime}}\right)^{p+\frac{d-1}{2}}$
$\displaystyle\times\sum_{m_{2}=0}^{p-m}\epsilon_{m_{2}}\cos(m_{2}(\phi_{2}-\phi_{2}^{\prime}))\cdots\sum_{m_{k}=0}^{p-\sum_{i=1}^{k-1}m_{i}}\cdots\sum_{m_{d/2}=0}^{p-\sum_{i=1}^{{d/2}-1}m_{i}}\epsilon_{m_{d/2}}\cos(m_{d/2}(\phi_{d/2}-\phi_{d/2}^{\prime}))$
$\displaystyle\times(-1)^{M}\sum_{n_{{d/2}-1}=0}^{\left\lfloor\frac{p-M}{2}\right\rfloor}\Psi_{{d/2}-1}^{\log_{2}d}\left(\begin{array}[]{c}{n_{{d/2}-1}}\\\
{m_{2},m}\end{array};\begin{array}[]{c}{\vartheta_{{d/2}-1}}\\\\[5.0pt]
{\vartheta_{{d/2}-1}^{\prime}}\end{array}\right)\cdots\sum_{n=0}^{\left\lfloor\frac{p-M-2\sum_{i=2}^{{d/2}-1}n_{i}}{2}\right\rfloor}\Psi_{1}^{\log_{2}d}\left(\begin{array}[]{c}{n}\\\
{l_{2},l_{3}}\end{array};\begin{array}[]{c}{\vartheta}\\\
{\vartheta^{\prime}}\end{array}\right)$ (7.47)
$\displaystyle\times\frac{1}{(p-2N-M)!(p+2N+M+d-2)!}Q_{2N+M+\frac{d-3}{2}}^{p+\frac{d-1}{2}}(\zeta).$
(7.48)

######  Proof 7.7.

Start with the Gegenbauer expansion of a fundamental solution given by Theorem
5.4. The hyperspherical harmonics contained within can be expanded with the
aid of (6.1) and Theorem 6.1 to be

$\displaystyle
C_{l}^{d/2-1}(\cos\gamma)=\frac{(d-2)}{(2l+d-2)\Gamma(d/2)}\sum_{K}\epsilon_{m}\cos(m(\phi-\phi^{\prime}))\epsilon_{m_{2}}\cos(m_{2}(\phi_{m_{2}}-\phi_{m_{2}}^{\prime}))\cdots\epsilon_{m_{d/2}}\cos(m_{d/2}(\phi_{d/2}-\phi_{d/2}^{\prime}))$
$\displaystyle\hskip
85.35826pt\times\Psi_{{d/2}-1}^{\log_{2}d}\left(\begin{array}[]{c}{n_{{d/2}-1}}\\\
{m_{2},m}\end{array};\begin{array}[]{c}{\vartheta_{{d/2}-1}}\\\\[5.0pt]
{\vartheta_{{d/2}-1}^{\prime}}\end{array}\right)\times\cdots\times\Psi_{1}^{\log_{2}d}\left(\begin{array}[]{c}{n}\\\
{l_{2},l_{3}}\end{array};\begin{array}[]{c}{\vartheta}\\\
{\vartheta^{\prime}}\end{array}\right).$ (7.57)

The multi-sum in (7.7) can be combined with the sum in (5.4), and with the
ability to reverse this multi-sum given by Lemma 1 yields:

$\displaystyle{\mathfrak{h}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})=\frac{2^{\frac{d}{2}-\frac{1}{2}}}{\sqrt{\pi}}{\mathrm{e}}^{i\pi(p-\frac{d}{2}+\frac{1}{2})}p!(2rr)^{p}\left(\frac{r_{>}^{2}-r_{<}^{2}}{2rr^{\prime}}\right)^{p+\frac{d-1}{2}}\sum_{m=0}^{p-m}\epsilon_{m}\cos(m(\phi-\phi^{\prime}))$
$\displaystyle\hskip
14.22636pt\times\sum_{m_{2}=0}^{p-m}\epsilon_{m_{2}}\cos(m_{2}(\phi_{2}-\phi_{2}^{\prime}))\cdots\sum_{m_{k}=0}^{p-\sum_{i=1}^{k-1}m_{i}}\cdots\sum_{m_{d/2}=0}^{p-\sum_{i=1}^{{d/2}-1}m_{i}}\epsilon_{m_{d/2}}\cos(m_{d/2}(\phi_{d/2}-\phi_{d/2}^{\prime}))$
$\displaystyle\hskip
42.67912pt\times(-1)^{M}\sum_{n_{{d/2}-1}=0}^{\left\lfloor\frac{p-M}{2}\right\rfloor}\Psi_{{d/2}-1}^{\log_{2}d}\left(\begin{array}[]{c}{n_{{d/2}-1}}\\\
{m_{2},m}\end{array};\begin{array}[]{c}{\vartheta_{{d/2}-1}}\\\\[5.0pt]
{\vartheta_{{d/2}-1}^{\prime}}\end{array}\right)\cdots\sum_{n=0}^{\left\lfloor\frac{p-M-2\sum_{i=2}^{{d/2}-1}n_{i}}{2}\right\rfloor}\Psi_{1}^{\log_{2}d}\left(\begin{array}[]{c}{n}\\\
{l_{2},l_{3}}\end{array};\begin{array}[]{c}{\vartheta}\\\
{\vartheta^{\prime}}\end{array}\right)$ (7.66) $\displaystyle\hskip
42.67912pt\times\frac{1}{(p-2N-M)!(2N+M+p+d-2)!}Q_{l+\frac{d-3}{2}}^{p+\frac{d-1}{2}}(\zeta).$
(7.67)

The Fourier coefficients of this can be compared with the Fourier expansion of
a fundamental solution given in Theorem 5.3. This completes the proof

The simplest example of a binomial addition theorem for $0\leq m\leq p$ in
generalized Hopf coordinates occurs in four dimensions.

######  Theorem 5.

Let $p,m\in{\mathbb{N}}_{0}$, $0\leq m\leq p$, $p=k-d/2\in{\mathbb{N}}_{0}$
and $d=4$. Then

$\displaystyle
Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)=-2(p-m)!(p+m)!(\chi^{2}-1)^{-\frac{p}{2}-\frac{1}{4}}\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}\left(\frac{r_{>}^{2}-r_{<}^{2}}{2rr^{\prime}}\right)^{p+\frac{3}{2}}(\sin\vartheta\sin\vartheta^{\prime})^{m}$
$\displaystyle\times\sum_{m_{2}=0}^{p-m}\epsilon_{m_{2}}\cos(m_{2}(\phi_{2}-\phi_{2}^{\prime}))(-1)^{m_{2}}(\cos\vartheta\cos\vartheta^{\prime})^{m_{2}}\sum_{n=0}^{\left\lfloor\frac{p-m-
m_{2}}{2}\right\rfloor}P_{n}^{(m,m_{2})}(\cos(2\vartheta))P_{n}^{(m,m_{2})}(\cos(2\vartheta^{\prime}))$
$\displaystyle\times\frac{(2n+m+m_{2}+1)(n+m+m_{2})!n!}{(p-2n-m-
m_{2})!(p+2n+m+m_{2}+2)!(n+m)!(n+m_{2})!}Q_{2n+m+m_{2}+\frac{1}{2}}^{p+\frac{3}{2}}(\zeta).$
(7.68)

######  Proof 7.8.

Starting with Theorem 7.6, let $d=4$. From this we obtain the following:

$\displaystyle
Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)=-2(p-m)!(p+m)!(\chi^{2}-1)^{-\frac{p}{2}-\frac{1}{4}}\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}\left(\frac{r_{>}^{2}-r_{<}^{2}}{2rr^{\prime}}\right)^{p+\frac{3}{2}}$
$\displaystyle\times\sum_{m_{2}=0}^{p-m}(-1)^{m_{2}}\epsilon_{m_{2}}\cos(m_{2}(\phi_{2}-\phi_{2}^{\prime}))\sum_{n=0}^{\left\lfloor\frac{p-m-
m_{2}}{2}\right\rfloor}\Psi_{1}^{2}\left(\begin{array}[]{c}{n}\\\
{m,m_{2}}\end{array};\begin{array}[]{c}{\vartheta}\\\\[5.0pt]
{\vartheta^{\prime}}\end{array}\right)$ (7.73) $\displaystyle\hskip
14.22636pt\times\frac{1}{(p-2n-m-
m_{2})!(p+2n+m+m_{2}+2)!}Q_{2n+m+m_{2}+\frac{1}{2}}^{p+\frac{3}{2}}(\zeta).$
(7.74)

Then by (6.17), (6.19), the function denoted by $\Psi$ can be written in terms
of Jacobi polynomials:

$\displaystyle\Psi_{1}^{2}\left(\begin{array}[]{c}{n}\\\
{m,m_{2}}\end{array};\begin{array}[]{c}{\vartheta}\\\\[5.0pt]
{\vartheta^{\prime}}\end{array}\right)=\frac{(2n+m+m_{2}+1)(n+m+m_{2})!n!}{(n+m)!(n+m_{2})!}(\sin\vartheta\sin\vartheta^{\prime})^{m}(\cos\vartheta\cos\vartheta^{\prime})^{m_{2}}$
(7.79) $\displaystyle\hskip 99.58464pt\times
P_{n}^{(m,m_{2})}(\cos(2\vartheta))P_{n}^{(m,m_{2})}(\cos(2\vartheta^{\prime})).$
(7.80)

Using this identity with (7.8) completes the proof.

In generalized Hopf coordinates, one has the following logarithmic addition
theorem valid for $0\leq m\leq p$.

######  Theorem 5.

Let $0\leq m\leq p$, $d=2^{q}$, $q\geq 2$ and $p=k-d/2\in{\mathbb{N}}_{0}$.
Then

$\displaystyle\frac{1}{(p+m)!(p-m)!}\left(\log(RR^{\prime})+\log(\chi+\sqrt{\chi^{2}-1})+2H_{2p}-H_{p+m}-H_{p-m}\right)Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)$
$\displaystyle+\frac{1}{(p+m)!}\sum_{k=0}^{p-m-1}\frac{(2m+2k+1)}{k!(p-m-k)(p+m+k+1)}\left[1+\frac{k!(p+m)!}{(2m+k)!(p-m)!}\right]Q_{m-\frac{1}{2}}^{m+k+\frac{1}{2}}(\chi)$
$\displaystyle+\frac{1}{(p-m)!}\sum_{k=0}^{m-1}\frac{2k+1}{(m+k)!(p-k)(p+k+1)}Q_{m-\frac{1}{2}}^{k+\frac{1}{2}}(\chi)$
$\displaystyle=(-1)^{m+{\frac{d}{2}}-1}2^{{\frac{d}{2}}-1}\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}(\chi^{2}-1)^{-\frac{p}{2}-\frac{1}{4}}\left(\frac{r_{>}^{2}-r_{<}^{2}}{2rr^{\prime}}\right)^{p+\frac{d-1}{2}}$
$\displaystyle\times\Biggl{\\{}\
\sum_{m_{2}=0}^{p-m}\epsilon_{m_{2}}\cos(m_{2}(\phi_{2}-\phi_{2}^{\prime}))\cdots\sum_{m_{j}=0}^{p-\sum_{i=1}^{j-1}m_{i}}\epsilon_{m_{j}}\cos(m_{j}(\phi_{j}-\phi_{j}^{\prime}))\cdots\sum_{m_{\frac{d}{2}}=0}^{p-\sum_{i=1}^{{\frac{d}{2}}-1}m_{i}}\epsilon_{m_{\frac{d}{2}}}\cos(m_{\frac{d}{2}}(\phi_{\frac{d}{2}}-\phi_{\frac{d}{2}}^{\prime}))$
$\displaystyle\times(-1)^{M}\sum_{n_{{\frac{d}{2}}-1}=0}^{\lfloor\frac{p-M}{2}\rfloor}\Psi_{{\frac{d}{2}}-1}^{\log_{2}d}\left(\begin{array}[]{c}{n_{{\frac{d}{2}}-1}}\\\
{m_{2},m}\end{array};\begin{array}[]{c}{\vartheta_{{\frac{d}{2}}-1}}\\\\[5.0pt]
{\vartheta_{{\frac{d}{2}}-1}^{\prime}}\end{array}\right)\cdots\sum_{n=0}^{\lfloor\frac{p+2n-2N-M}{2}\rfloor}\Psi_{1}^{\log_{2}d}\left(\begin{array}[]{c}{n}\\\
{l_{2},l_{3}}\end{array};\begin{array}[]{c}{\vartheta}\\\
{\vartheta^{\prime}}\end{array}\right)$ (7.89) $\displaystyle\times\biggr{(}\
\frac{2\log
r_{>}+2H_{2p+d-2}+H_{p}-H_{p+{\frac{d}{2}}-1}-H_{p+M+2N+d-2}-H_{p-2N-M}}{(p-2N-M)!(p+2N+M+d-2)!}Q_{2N+M+\frac{d-3}{2}}^{p+\frac{d-1}{2}}(\zeta)$
$\displaystyle\hskip
14.22636pt+\frac{1}{(p+2N+M+d-2)!}\sum_{k=0}^{p-2N-M-1}\frac{(4N+2M+2k+d-1)}{k!(p-k-2N-M)(p+k+2N+M+d-1)}$
$\displaystyle\hskip
28.45274pt\times\left[1+\frac{k!(p+2N+M+d-2)!}{(k+4N+2M+d-2)!(p-2N-M)!}\right]Q_{2N+M+\frac{d-3}{2}}^{k+2N+M+\frac{d-1}{2}}(\zeta)$
$\displaystyle\hskip
14.22636pt+\frac{1}{(p-2N-M)!}\sum_{k=0}^{2N+M+{\frac{d}{2}}-2}\frac{2k+1}{(2N+M+k+{\frac{d}{2}}-1)!(p+{\frac{d}{2}}-k-1)(p+k+{\frac{d}{2}})}Q_{2N+M+\frac{d-3}{2}}^{k+\frac{1}{2}}(\zeta)\
\biggr{)}$
$\displaystyle+(-1)^{p+1}\sum_{m_{2}=0}^{\infty}\epsilon_{m_{2}}\cos(m_{2}(\phi_{2}-\phi_{2}^{\prime}))\cdots\sum_{m_{\frac{d}{2}}=0}^{\infty}\epsilon_{m_{\frac{d}{2}}}\cos(m_{\frac{d}{2}}(\phi_{\frac{d}{2}}-\phi_{\frac{d}{2}}^{\prime}))$
$\displaystyle\hskip
14.22636pt\times\sum_{n_{{\frac{d}{2}}-1}=0}^{\infty}\Psi_{{\frac{d}{2}}-1}^{\log_{2}d}\left(\begin{array}[]{c}{n_{{\frac{d}{2}}-1}}\\\
{m_{2},m}\end{array};\begin{array}[]{c}{\vartheta_{{\frac{d}{2}}-1}}\\\\[5.0pt]
{\vartheta_{{\frac{d}{2}}-1}^{\prime}}\end{array}\right)\cdots\sum_{n_{2}=0}^{\infty}\Psi_{2}^{\log_{2}d}\left(\begin{array}[]{c}{n_{2}}\\\
{l_{4},l_{5}}\end{array};\begin{array}[]{c}{\vartheta_{2}}\\\\[5.0pt]
{\vartheta_{2}^{\prime}}\end{array}\right)$ (7.98) $\displaystyle\hskip
14.22636pt\times\sum_{n=\max\left(0,\lfloor\frac{p-2N-M}{2}\rfloor+1\right)}^{\infty}\frac{(2N+M-p-1)!}{(2N+M+p+d-2)!}\Psi_{1}^{\log_{2}d}\left(\begin{array}[]{c}{n}\\\
{l_{2},l_{3}}\end{array};\begin{array}[]{c}{\vartheta}\\\
{\vartheta^{\prime}}\end{array}\right)Q_{2N+M+\frac{d-3}{2}}^{p+\frac{d-1}{2}}(\zeta)\Biggr{\\}}.$
(7.103)

######  Proof 7.9.

Again, we start by comparing the Fourier and Gegenbauer expansions of the
logarithmic fundamental solutions. For the Gegenbauer expansion, replace $\mu$
with $\frac{d}{2}-1$, use (3.13) and the following relation:

$\log(rr^{\prime})+\log\left(\zeta+\sqrt{\zeta^{2}-1}\right)=2\log r_{>}.$

The Gegenbauer expansion of Theorem 5.2 can be rewritten in Hopf coordinates
by use of (7.7). Next, we consider the multi-sums. The first sums of
$\sum_{l=0}^{p}\sum_{K}$, are dealt with by Lemma 1. The later sum uses Lemma
2. Using the identity $l=2N+M$ and compare the Fourier coefficients to
complete the proof.

The simplest example of a logarithmic addition theorem for $0\leq m\leq p$ in
generalized Hopf coordinates occurs in four dimensions.

######  Corollary 6.

Let $0\leq m\leq p$, $m\in{\mathbb{N}}_{0}$, $p=k-d/2\in{\mathbb{N}}_{0}$
where $d=4$. Then

$\displaystyle\frac{1}{(p+m)!(p-m)!}\left(\log(RR^{\prime})+\log(\chi+\sqrt{\chi^{2}-1})+2H_{2p}-H_{p+m}-H_{p-m}\right)Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)$
$\displaystyle+\frac{1}{(p+m)!}\sum_{k=0}^{p-m-1}\frac{(2m+2k+1)}{k!(p-m-k)(p+m+k+1)}\left[1+\frac{k!(p+m)!}{(2m+k)!(p-m)!}\right]Q_{m-\frac{1}{2}}^{m+k+\frac{1}{2}}(\chi)$
$\displaystyle+\frac{1}{(p-m)!}\sum_{k=0}^{m-1}\frac{2k+1}{(m+k)!(p-k)(p+k+1)}Q_{m-\frac{1}{2}}^{k+\frac{1}{2}}(\chi)$
$\displaystyle=-2\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}(\chi^{2}-1)^{-\frac{p}{2}-\frac{1}{4}}\left(\frac{r_{>}^{2}-r_{<}^{2}}{2rr^{\prime}}\right)^{p+\frac{3}{2}}(\sin\vartheta\sin\vartheta^{\prime})^{m}$
$\displaystyle\times\Biggl{\\{}\
\sum_{m_{2}=0}^{p-m}\epsilon_{m_{2}}\cos(m_{2}(\phi_{2}-\phi_{2}^{\prime}))(\cos\vartheta\cos\vartheta^{\prime})^{m_{2}}(-1)^{m_{2}}$
$\displaystyle\times\sum_{n=0}^{\lfloor\frac{p-m-
m_{2}}{2}\rfloor}\frac{(2n+m+m_{2}+1)(n+m+m_{2})!n!}{(n+m)!(n+m_{2})!}P_{n}^{(m,m_{2})}(\cos(2\vartheta))P_{n}^{(m,m_{2})}(\cos(2\vartheta^{\prime}))$
$\displaystyle\times\biggr{(}\ \frac{2\log
r_{>}+2H_{2p+2}+H_{p}-H_{p+1}-H_{p+2n+m+m_{2}+2}-H_{p-2n-m-m_{2}}}{(p-2n-m-
m_{2})!(p+2n+m+m_{2}+2)!}Q_{2n+m+m_{2}+\frac{1}{2}}^{p+\frac{3}{2}}(\zeta)$
$\displaystyle\hskip 14.22636pt+\frac{1}{(p+2n+m+m_{2}+2)!}\sum_{k=0}^{p-2n-m-
m_{2}-1}\frac{(4n+2m+2m_{2}+2k+3)}{k!(p-k-2n-m-m_{2})(p+k+2n+m+m_{2}+3)}$
$\displaystyle\hskip
28.45274pt\times\left[1+\frac{k!(p+2n+m+m_{2}+2)!}{(k+4n+2m+2m_{2}+2)!(p-2n-m-
m_{2})!}\right]Q_{2n+m+m_{2}+\frac{1}{2}}^{k+2n+m+m_{2}+\frac{3}{2}}(\zeta)$
$\displaystyle\hskip 14.22636pt+\frac{1}{(p-2n-m-
m_{2})!}\sum_{k=0}^{2n+m+m_{2}}\frac{2k+1}{(2n+m+m_{2}+k+1)!(p-k+1)(p+k+2)}Q_{2n+m+m_{2}+\frac{1}{2}}^{k+\frac{1}{2}}(\zeta)\
\biggr{)}$
$\displaystyle+(-1)^{p+m+1}\sum_{m_{2}=0}^{\infty}\epsilon_{m_{2}}\cos(m_{2}(\phi_{2}-\phi_{2}^{\prime}))(\cos\vartheta\cos\vartheta^{\prime})^{m_{2}}$
$\displaystyle\hskip 14.22636pt\times\sum_{n=\max\left(0,\lfloor\frac{p-m-
m_{2}}{2}\rfloor+1\right)}^{\infty}\frac{(2n+m+m_{2}+1)(n+m+m_{2})!(2n+m+m_{2}-p-1)!n!}{(n+m)!(n+m_{2})!(2n+m+m_{2}+p+2)!}$
$\displaystyle\hskip 113.81102pt\times
P_{n}^{(m,m_{2})}(\cos(2\vartheta))P_{n}^{(m,m_{2})}(\cos(2\vartheta^{\prime}))Q_{2n+m+m_{2}+\frac{1}{2}}^{p+\frac{3}{2}}(\zeta)\Biggr{\\}}.$

######  Proof 7.10.

When $d=4$, note that $N=n$ and $M=m+m_{2}$. The functions of the Hopf
coordinate system can be simplified as in Theorem 7.7.

The general logarithmic addition theorem for $m\geq p+1$ in generalized Hopf
coordinates is given as follows.

######  Theorem 6.

Let $p,m\in{\mathbb{N}}_{0}$, $m\geq p+1$, $d=2^{q}$, $q\geq 2$. Then

$\displaystyle
Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)=(-2)^{{\frac{d}{2}}-1}2^{{\frac{d}{2}}-1}\frac{(p+m)!}{(m-p-1)!}(\chi^{2}-1)^{-\frac{p}{2}-\frac{1}{4}}\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}\left(\frac{r_{>}^{2}-r_{<}^{2}}{2rr^{\prime}}\right)^{p+\frac{d-1}{2}}$
$\displaystyle\times\sum_{m_{2}=0}^{\infty}\epsilon_{m_{2}}\cos(m_{2}(\phi_{2}-\phi_{2}^{\prime}))\times\cdots\times\sum_{m_{\frac{d}{2}}=0}^{\infty}\epsilon_{m_{\frac{d}{2}}}\cos(m_{\frac{d}{2}}(\phi_{\frac{d}{2}}-\phi_{\frac{d}{2}}^{\prime}))\sum_{n_{{\frac{d}{2}}-1}=0}^{\infty}\Psi_{{\frac{d}{2}}-1}^{\log_{2}d}\left(\begin{array}[]{c}{n_{{\frac{d}{2}}-1}}\\\
{m_{2},m}\end{array};\begin{array}[]{c}{\vartheta_{{\frac{d}{2}}-1}}\\\\[5.0pt]
{\vartheta_{{\frac{d}{2}}-1}^{\prime}}\end{array}\right)$ (7.108)
$\displaystyle\times\cdots\times\sum_{n=0}^{\infty}\Psi_{1}^{\log_{2}d}\left(\begin{array}[]{c}{n}\\\
{l_{2},l_{3}}\end{array};\begin{array}[]{c}{\vartheta}\\\
{\vartheta^{\prime}}\end{array}\right)\frac{(2N+M-p-1)!}{(p+2N+M+d-2)!}Q_{2N+M+\frac{d-3}{2}}^{p+\frac{d-1}{2}}(\zeta).$
(7.113)

######  Proof 7.11.

Similar to Theorem 7.8. For this case only the Fourier coefficients where
$m\geq p+1$ are used, therefore Lemma 3 is needed. Simplifying the expression
completes the proof.

The simplest example of a logarithmic addition theorem for $m\geq p+1$ in
generalized Hopf coordinates occurs in four dimensions.

######  Theorem 6.

Let $p,m\in{\mathbb{N}}_{0}$, $m\geq p+1$. Then

$\displaystyle
Q_{m-\frac{1}{2}}^{p+\frac{1}{2}}(\chi)=-2\frac{(p+m)!}{(m-p-1)!}(\chi^{2}-1)^{-\frac{p}{2}-\frac{1}{4}}\left(\frac{rr^{\prime}}{RR^{\prime}}\right)^{p}\left(\frac{r_{>}^{2}-r_{<}^{2}}{2rr^{\prime}}\right)^{p+\frac{3}{2}}(\sin\vartheta\sin\vartheta^{\prime})^{m}$
$\displaystyle\times\sum_{m_{2}=0}^{\infty}\epsilon_{m_{2}}\cos(m_{2}(\phi_{2}-\phi_{2}^{\prime}))(\cos\vartheta\cos\vartheta^{\prime})^{m_{2}}\sum_{n=0}^{\infty}P_{n}^{(m,m_{2})}(\cos(2\vartheta))P_{n}^{(m,m_{2})}(\cos(2\vartheta^{\prime}))$
$\displaystyle\times\frac{(2n+m+m_{2}+1)(2n+m+m_{2}-p-1)!(n+m+m_{2})!n!}{(p+2n+m+m_{{2}}+2)!(n+m)!(n+m_{2})!}Q_{2n+m+m_{2}+\frac{1}{2}}^{p+\frac{3}{2}}(\zeta).$
(7.114)

######  Proof 7.12.

Simplifying as in Theorem 7.8 for $d=4$ completes the proof.

### Acknowledgements

We would like to thank A.F.M. Tom ter Elst, Matthew Auger and Radosław
Szmytkowski for valuable discussions.

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