Matrix valued orthogonal polynomials of the Jacobi type (Q1432897)

This paper describes a method for the construction of families of orthogonal polynomials having matrices as coefficients. The construction is based on the definition of a classical pair \((W, D)\). Here \(W\) is a self-adjoint, positive definite, matrix valued weight function; \(D\) is a second order ordinary differential operator with matrix valued polynomial coefficients and \(D\) satisfies a symmetry condition with respect to a matrix valued pairing determined by \(W\). The explicit construction of three families of orthogonal polynomials \(P_n\) is given when \(D\) is of hypergeometric type. In these cases, the polynomials \(P_n\) can be expressed in terms of matrix valued hypergeometric functions as those introduced in [\textit{J. A. Tirao}, Proc. Nat. Acad. Sci. USA 100, No. 14, 8138--8141 (2003; Zbl 1070.33015)].