Weighted $L^2$-Norms of Gegenbauer polynomials

TL;DR

This paper derives exact, generating, and asymptotic formulas for weighted L2 norms of Gegenbauer polynomials, enhancing understanding of their integral properties with specific weight functions.

Contribution

It provides new exact and asymptotic formulas for integrals of squared Gegenbauer polynomials with weights, expanding analytical tools for these functions.

Findings

Exact formulas for weighted L2 norms of Gegenbauer polynomials.

Generating functions for these integrals.

Asymptotic behavior as polynomial degree increases.

Abstract

We study integrals of the form \begin{equation*} \int_{-1}^1(C_n^{(\lambda)}(x))^2(1-x)^\alpha (1+x)^\beta\, dx, \end{equation*} where $C_n^{(\lambda)}$ denotes the Gegenbauer-polynomial of index $\lambda>0$ and $\alpha,\beta>-1$. We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as $n\to\infty$.