On hyperspherical associated Legendre functions: the extension of spherical harmonics to $N$ dimensions

TL;DR

This paper extends spherical harmonics to N dimensions by introducing hyperspherical associated Legendre functions, providing solutions for various PDEs in hyperspherical coordinates, and showing their relation to hypergeometric functions.

Contribution

It introduces hyperspherical associated Legendre functions as a generalization of associated Legendre functions for N-dimensional problems, connecting them to hypergeometric functions.

Findings

Derived solutions for PDEs in N-dimensional hyperspherical coordinates

Extended the class of Legendre functions to hyperspherical associated Legendre functions

Established the relation to Gaussian hypergeometric functions

Abstract

The solution in hyperspherical coordinates for $N$ dimensions is given for a general class of partial differential equations of mathematical physics including the Laplace, wave, heat and Helmholtz, Schr\"{o}dinger, Klein-Gordon and telegraph equations and their combinations. The starting point is the Laplacian operator specified by the scale factors of hyperspherical coordinates. The general equation of mathematical physics is solved by separation of variables leading to the dependencies: (i) on time by the usual exponential function; (ii) on longitude by the usual sinusoidal function; (iii) on radius by Bessel functions of order generally distinct from cylindrical or spherical Bessel functions; (iv) on one latitude by associated Legendre functions; (v) on the remaining latitudes by an extension, namely the hyperspherical associated Legendre functions. The original associated Legendre functions are a particular case of the Gaussian hypergeometric functions, and the hyperspherical associated Legendre functions are also a more general particular case of the Gaussian hypergeometric functions so that it is not necessary to consider extended Gaussian hypergeometric functions.