Linearization coefficients for some orthogonal polynomials

Michael Anshelevich

UC Riverside

February 26,
refreshments at 3:45pm


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 q-interpolated 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 partition-dependent stochastic measures of Rota and Wallstrom.

No prior knowledge of the material will be assumed.

Speaker's Contact Info: manshel(at-sign)

Return to seminar home page

Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)

Page loaded on February 03, 2003 at 05:02 PM. Copyright © 1998-99, Sara C. Billey. All rights reserved.