Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems

We develop complex Jacobi, Gegenbauer and Chebyshev polynomial expansions for the kernels associated with power-law fundamental solutions of the polyharmonic equation on d-dimensional Euclidean space. From these series representations we derive Fourier expansions in certain rotationally-invariant coordinate systems and Gegenbauer polynomial expansions in Vilenkin's polyspherical coordinates. We compare both of these expansions to generate addition theorems for the azimuthal Fourier coefficients

Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

Due to the isotropy of $d$-dimensional hyperspherical space, one expects there to exist a spherically symmetric opposite antipodal fundamental solution for its corresponding Laplace-Beltrami operator. The $R$-radius hypersphere ${\mathbf S}_R^d$ with $R>0$, represents a Riemannian manifold with positive-constant sectional curvature. We obtain a spherically symmetric opposite antipodal fundamental solution of Laplace's equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the trigonometric sine, finite summation expressions over trigonometric functions, Gauss hypergeometric functions, and in terms of the Ferrers function of the second with degree and order given by $d/2-1$ and $1-d/2$ respectively, with real argument $x\in(-1,1)$.Comment: essentially corrected versio

Fourier expansions for a logarithmic fundamental solution of the polyharmonic equation

In even-dimensional Euclidean space for integer powers of the Laplacian greater than or equal to the dimension divided by two, a fundamental solution for the polyharmonic equation has logarithmic behavior. We give two approaches for developing a Fourier expansion of this logarithmic fundamental solution. The first approach is algebraic and relies upon the construction of two-parameter polynomials. We describe some of the properties of these polynomials, and use them to derive the Fourier expansion for a logarithmic fundamental solution of the polyharmonic equation. The second approach depends on the computation of parameter derivatives of Fourier series for a power-law fundamental solution of the polyharmonic equation. The resulting Fourier series is given in terms of sums over associated Legendre functions of the first kind. We conclude by comparing the two approaches and giving the azimuthal Fourier series for a logarithmic fundamental solution of the polyharmonic equation in rotationally-invariant coordinate systems

Generalized Heine Identity for Complex Fourier Series of Binomials

In this paper we generalize an identity first given by Heinrich Eduard Heine in his treatise, {\it Handbuch der Kugelfunctionen, Theorie und Anwendungen (1881), which gives a Fourier series for $1/[z-\cos\psi]^{1/2}$, for $z,\psi\in\R$, and $z>1$, in terms of associated Legendre functions of the second kind with odd-half-integer degree and vanishing order. In this paper we give a generalization of this identity as a Fourier series of $1/[z-\cos\psi]^\mu$, where z,\mu\in\C, $|z|>1$, and the coefficients of the expansion are given in terms of the same functions with order given by $\frac12-\mu$. We are also able to compute certain closed-form expressions for associated Legendre functions of the second kind.Comment: 12 page

Eigenfunction expansions for a fundamental solution of Laplace's equation on R3\R^3 in parabolic and elliptic cylinder coordinates

A fundamental solution of Laplace's equation in three dimensions is expanded in harmonic functions that are separated in parabolic or elliptic cylinder coordinates. There are two expansions in each case which reduce to expansions of the Bessel functions $J_0(kr)$ or $K_0(kr)$, $r^2=(x-x_0)^2+(y-y_0)^2$, in parabolic and elliptic cylinder harmonics. Advantage is taken of the fact that $K_0(kr)$ is a fundamental solution and $J_0(kr)$ is the Riemann function of partial differential equations on the Euclidean plane