Elementary Examples of Solutions to Bochner's Problem for Matrix Differential Operators

Abstract

In this paper, we demonstrate an elementary method for constructing new solutions to Bochner's problem for matrix differential operators from known solutions. We then describe a large family of solutions to Bochner's problem, obtained from classical solutions, which include several examples known from the literature. By virtue of the method of construction, we show how one may explicitly identify a generating function for the associated sequence of monic w-orthogonal matrix polynomials p(x,n) as well as the associated algebra D(w) of all matrix differential operators for which the p(x,n) are eigenfunctions. We also include some general results on the structure of the algebra D(w).

38 pages; expanded examples to higher dimensions; fixed issues with the previous definition of adjoint; modified main theorem to a simpler and cleaner version; rewrote introduction to be friendlier and more self-contained; added many additional references; fixed typos; added more general results about the structure of D(w)