On the $L^2$-norm of Gegenbauer polynomials

TL;DR

This paper derives a recursive formula and analyzes the asymptotic behavior of the $L^2$-norm of Gegenbauer polynomials, which are important in numerical analysis and interpolation.

Contribution

It provides the first recursive formula and asymptotic analysis for the $L^2$-norm of Gegenbauer polynomials, enhancing understanding of their properties.

Findings

Recursive formula for the $L^2$-norm derived

Asymptotic behavior characterized

Implications for numerical analysis discussed

Abstract

Gegenbauer, also known as ultra-spherical polynomials appear often in numerical analysis or interpolation. In the present text we find a recursive formula and compute the asymptotic behavior for their $L^2$-norm.