Kac's kinetic theory model and Gasper's identity

In 1954 Mark Kac proposed a collision model for N particles that would indicate how a physical system could evolve to equilibrium as the number of particles tends to infinity. He postulated but was unable to prove that there would be a gap in the energy spectrum and this would show the decay to the equilibrium state. In their resolution of this problem Carlen, Carvalho, and Loss introduced a family of correlation operators which were decisive in their analysis. A remarkable fact is that these operators allow simple proofs of Gegenbauer's identity for products of ultraspherical polynomials and Gasper's positivity results on sums of triple products of Jacobi polynomials. Gasper's results generalized those of Bochner who earlier proved a similar result for ultraspherical polynomials. I will discuss the Kac model and its connection to Bochner's, Gasper's and Gegenbauer's results. This is joint work with Eric Carlen and Michael Loss.