An integral identity with applications in orthogonal polynomials

TL;DR

This paper establishes a new integral identity involving parameters and applies it to derive relations for Gegenbauer polynomials, leading to explicit formulas for reproducing kernels of orthogonal polynomials on standard domains.

Contribution

It introduces a novel integral identity for multivariate functions and uses it to obtain new relations for Gegenbauer polynomials and explicit reproducing kernels.

Findings

Derived a new integral relation for Gegenbauer polynomials.

Obtained closed-form formulas for reproducing kernels on the cube and ball.

Connected integral identities with orthogonal polynomial theory.

Abstract

For $\boldsymbol{\large {\lambda}} = (\lambda_1,\ldots,\lambda_d)$ with $\lambda_i > 0$, it is proved that \begin{equation*} \prod_{i=1}^d \frac{ 1}{(1- r x_i)^{\lambda_i}} = \frac{\Gamma(|\boldsymbol{\large {\lambda}}|)}{\prod_{i=1}^{d} \Gamma(\lambda_i)} \int_{\mathcal{T}^d} \frac{1}{ (1- r \langle x, u \rangle)^{|\boldsymbol{\large {\lambda}}|}} \prod_{i=1}^d u_i^{\lambda_i-1} du, \end{equation*} where $\mathcal{T}^d$ is the simplex in homogeneous coordinates of $\mathbb{R}^d$, from which a new integral relation for Gegenbuer polynomials of different indexes is deduced. The latter result is used to derive closed formulas for reproducing kernels of orthogonal polynomials on the unit cube and on the unit ball.