Some integrals and series involving the Gegenbauer polynomials and the Legendre functions on the cut (-1,1)

TL;DR

This paper evaluates integrals involving Gegenbauer polynomials and expresses the results using Legendre functions on the cut, also deriving closed-form series representations with applications in special functions analysis.

Contribution

It provides new integral evaluations and series representations involving Gegenbauer polynomials and Legendre functions, extending recent findings by Cohl.

Findings

Explicit integral formulas in terms of Legendre functions.

Closed-form series representations involving Gegenbauer and Legendre functions.

Conditions on parameters for the validity of the formulas.

Abstract

We use the recent findings of Cohl [arXiv:1105.2735] and evaluate two integrals involving the Gegenbauer polynomials: $\int_{-1}^{x}\mathrm{d}t\:(1-t^{2})^{\lambda-1/2}(x-t)^{-\kappa-1/2}C_{n}^{\lambda}(t)$ and $\int_{x}^{1}\mathrm{d}t\:(1-t^{2})^{\lambda-1/2}(t-x)^{-\kappa-1/2}C_{n}^{\lambda}(t)$, both with $\Real\lambda>-1/2$, $\Real\kappa<1/2$, $-1<x<1$. The results are expressed in terms of the on-the-cut associated Legendre functions $P_{n+\lambda-1/2}^{\kappa-\lambda}(\pm x)$ and $Q_{n+\lambda-1/2}^{\kappa-\lambda}(x)$. In addition, we find closed-form representations of the series $\sum_{n=0}^{\infty}(\pm)^{n}[(n+\lambda)/\lambda]P_{n+\lambda-1/2}^{\kappa-\lambda}(\pm x)C_{n}^{\lambda}(t)$ and $\sum_{n=0}^{\infty}(\pm)^{n}[(n+\lambda)/\lambda]Q_{n+\lambda-1/2}^{\kappa-\lambda}(\pm x)C_{n}^{\lambda}(t)$, both with $\Real\lambda>-1/2$, $\Real\kappa<1/2$, $-1<t<1$, $-1<x<1$.