A class of matrix-valued polynomials generalizing Jacobi polynomials

A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a two-step recurrence relation, integral inter-relations, and quasi-orthogonality relations. 1. Motivation The understanding of matrix-valued orthogonal polynomials has advanced greatly in recent years, and a wealth of references can be found in the recent work of Barry Simon [1]. The polynomials introduced in the present paper generalize the Jacobi polynomials. A different generalization, yielding an orthogonal class, is due to Grünbaum [2]. The polynomials defined here were not sought-for. Rather they appear naturally in the forefront of a different problem, the study of the possibility to linearize (by an analytic change of variables) a differential equation whose linear part has only regular singular points. In a neighborhood of one such point equations are (generically) linearizable [3]. But this is no longer the case if the domain studied contains two such singularities. For example: (1) du dx = M u + f(x,u) with M = 1 x− 1 A+ 1 x+ 1 B (f collects all the nonlinear terms in u) is not necessarily equivalent to its linear part (2) dw dx = M w for x in a domain in the complex plane including both singular points ±1. It turns out, however that for any nonlinear term f(x,u) there exists a unique φ(u) so that the equation with the ”corrected” nonlinear part f(x,u)− φ(u) is linearizable [4]. Besides its clear intrinsic interest, the problem of detecting linearizable equations is important also since linearizability and integrability turn out to be intimately connected [5]. 1