13 - Some Families of Matrix-Valued Jacobi Orthogonal Polynomials

Summary

Among the classical (scalar-valued) families of orthogonal polynomials with rich and deep connections to several branches of mathematics, the Jacobi polynomials occupy a distinguished role.

In this contribution we describe a way of obtaining some families of matrix-valued orthogonal polynomials of arbitrary dimension and depending on two parameters α, β, which extends the scalar theory in many respects. We will achieve this goal by focusing on a group representation approach. In the scalar case the Jacobi polynomials appeared in several concrete mathematical physics problems in the hands of people like Laplace and Legendre. The group-theoretical interpretation, in the hands of E. Cartan and H. Weyl, is of more recent vintage.