On a generalization of the generating function for Gegenbauer polynomials

Published

January 28, 2013

Author(s)

Howard S. Cohl

Abstract

We derive a Gegenbauer polynomial expansion for complex powers of the distance between two points in $d$-dimensional Euclidean space. The argument of the Gegenbauer polynomial in the expansion is given by the cosine of the separation angle between the two points as measured from the origin. The order of the Gegenbauer polynomial is given by $d/2-1$, which is ideal for utilization of the addition theorem for hyperspherical harmonics. The coefficients of the expansion are given in terms of an associated Legendre function of the second kind with argument $(r^2+{r^\prime}^2)/(2rr^\prime)>1$, where $r,r^\prime$ represent the Euclidean norm of the vectors representing the distances to the two points as measured from the origin. We extend this result by proving a generalization of the generating function for Gegenbauer polynomials.

Citation

Integral Transforms and Special Functions

Pub Type

Journals

Download Paper

https://doi.org/10.1080/10652469.2012.761613

Keywords

Euclidean space, Polyharmonic equation, Fundamental solution, Gegenbauer polynomials, associated Legendre functions

Citation

Cohl, H. (2013), On a generalization of the generating function for Gegenbauer polynomials, Integral Transforms and Special Functions, [online], https://doi.org/10.1080/10652469.2012.761613 (Accessed October 22, 2025)