**
TALK BY MICHAEL ANSHELEVICH, UCR
**

**
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.