On the Markov sequence problem for Jacobi polynomials

TL;DR

This paper provides an elementary proof of Gasper's theorem on the Markov sequence problem for Jacobi polynomials, utilizing spectral analysis of an operator linked to a probabilistic molecular collision model, and introduces new integral formulas for Jacobi polynomial ratios.

Contribution

It offers a new, simple proof of Gasper's theorem and derives novel integral formulas for ratios of Jacobi polynomials, expanding existing mathematical tools.

Findings

Elementary proof of Gasper's theorem achieved

New integral formulas for Jacobi polynomial ratios derived

Connections established between spectral analysis and probabilistic models

Abstract

We give a simple and entirely elementary proof of Gasper's theorem on the Markov sequence problem for Jacobi polynomials. It is based on the spectral analysis of an operator that arises in the study of a probabilistic model of colliding molecules introduced by Marc Kac. In the process, we obtain some new integral formulas for ratios of Jacobi polynomials that generalize Gasper's product formula and a well known formula of Koornwinder.