Georgia Institute of Technology College of Sciences

Gasper in his 1971 Annals of Math paper proved that the Jacobi polynomials satisfy a product formula which generalized the product formula of Gegenbauer for ultraspherical polynomials. Gasper proved this by showing that certains sums of triple products of Jacobi polynomials are positive generalizing results of Bochner who earlier proved a similar results for ultraspherical polynomials. These results allow a convolution structure for Jacobi polynomials. We will give a simple proof of Gasper's and Bochner's results using a Markov operator found by Carlen, Carvahlo, and Loss in their study of the Kac model in kinetic theory. This is joint work with Eric Carlen and Michael Loss.