Linearization coefficients for some orthogonal polynomialsMichael AnshelevichUC Riverside
February 26,

ABSTRACT


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. After giving a brief introduction to the general theory of orthogonal polynomials, I will describe some families for which the linearization coefficients are known. These include Hermite, Charlier, Laguerre, Meixner, Chebyshev, and some qinterpolated families. In the second half of the talk, I will describe one method that can be used to calculate many of these coefficients. The method is only slightly combinatorial; instead, it is based on the partitiondependent stochastic measures of Rota and Wallstrom. No prior knowledge of the material will be assumed. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

