Some families of matrix valued Jacobi polynomials

F. A. GRUNBAUM, I. PACHARONI, J.TIRAO

Encyclopedia of Special Functions, The Askey Bateman Project, Volumen 1.

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Año: 2011;

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;  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. It is clear that in the matrix case the historical path is reversed and it remains an interesting challenge to nd good concrete applications of these families of matrix valued polynomials which satisfy three term recursions as well as dierential equations. In the very last section we refer to one recent use of the recursion relation satisfied by the matrix Jacobi polynomials. An approach that would attach a physical meaning to these matrix dierential equations could make these new special functions into a powerful tool in dierent areas.