Burchnall-type identities for matrix-valued orthogonal polynomials

PABLO ROMÁN

Tianjin

Congreso; Summer Research Institute on q-Series; 2018

Nankai University

We consider matrix-valued orthogonal polynomials with respect to a positive definite weight $W$ with finite moments. We discuss the existence of a suitable matrix-valued Pearson equation for $W$ which leads to explicit lowering and raising operators. This allows us to introduce a one parameter extension $W^{(u)}$, $u>0$, such that the lowering and raising operators relate the weights $W^{(u)}$ and $W^{(u+1)}$.We establish explicit matrix-valued Pearson equations for a family of Gegenbauer-type orthogonal polynomials related to the harmonic analysis of the group $mathrm{SU}(2)imes mathrm{SU}(2)$ and for families of Hermite-type and Laguerre-type matrix-valued orthogonal polynomials introduced by A. Dur´an and A. Gr"unbaum. For this we use an explicit LDU-decomposition in terms of classical orthogonal polynomials for these weights. Using the shift operators, we find the squared norm, the three-term recurrence relation and we establish a simple Rodrigues formula for these polynomials. We also use shift operators to obtain Burchnall-type identities for these families. We finally use these identities in order to give a non-trival solution to the non-abelian Toda lattice equations introduced by Gekhtman.An extension to a family of matrix-valued $q$-ultraspherical polynomials related to the quantum analogue of $mathrm{SU}(2)imes mathrm{SU}(2)$ will be discussed.This talk is based on joint papers with E. Koelink and M. Ismail.