On the Markov sequence problem for Jacobi polynomials

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.

The second version contains additional material and references. In particular, we discuss a product formula of Koornwider and Schwarz, and show how it may be proved using the methods developed here.