Linearization coefficients for orthogonal polynomials using stochastic processes
Michael Anshelevich, UC Riverside

A family of polynomials {P_n} such that P_n has degree n is a basis for the polynomial ring. A product P_{n_1} P_{n_2} ... P_{n_k} can be expanded in this basis, and the coefficients in this expansion are called linearization coefficients. If the basis consists of orthogonal polynomials, these coefficients are generalizations of the moments of the measure of orthogonality. Just like moments, these coefficients have combinatorial significance for many classical families. For instance, for the Hermite polynomials they are the numbers of inhomogeneous matchings. I will describe the linearization coefficients for a number of classical families. The proofs are based on the relation between the polynomials and certain stochastic processes. They involve the machinery of combinatorial stochastic measures, introduced by Rota and Wallstrom. The number of examples treated by this method is increased significantly by using non-commutative stochastic processes, consisting of operators on a q-deformed full Fock space.