Integral Representations of Ultraspherical Polynomials II

TL;DR

This paper improves the numerical stability of integral representations of ultraspherical polynomials by summing an infinite series into a single integral, enhancing computational practicality.

Contribution

It provides a numerically stable integral representation of ultraspherical polynomials of higher index, building on previous theoretical series representations.

Findings

Derived a numerically stable integral form for ultraspherical polynomials

Enhanced computational efficiency for high-degree polynomial evaluations

Validated the new integral representation through numerical experiments

Abstract

In the first part, by the first author's work of 1972, an integral representation for an ultraspherical polynomial of higher index in terms of one of lower index and an infinite series was obtained. While this representation works well from a theoretical point of view, it is not numerically satisfactory as it involves polynomials of high degree, which are numerically unstable. Here we sum this series to obtain an integral, which is numerically tractable.