Infinite Matrices in the Theory of Orthogonal Polynomials

Abstract

We obtain some properties of a class of q-hypergeometric orthogonal polynomials with q = −1, described by a uniform parametrization of the recurrence coefficients. We show that our class contains the Bannai-Ito polynomials and other known -1 polynomials. We introduce some new examples of -1 polynomials and also obtain matrix realizations of the Bannai-Ito algebra.

On classical orthogonal polynomials and the Cholesky factorization of a class of Hankel matrices

“…The bibliography on this subject has grown greatly in the last years, and it has become increasingly difficult to do a comprehensive review of all the references. Hence, we refer the reader to [22,23] (and the refereces therein) where the algebra of infinite triangular matrices and the algebra of infinite Hessenberg matrices are used to study some aspects of orthogonal polynomials, and to [5,15] (and the references therein) where the main tool is the Cholesky factorization of Gram matrices of bilinear forms. We remark that the Cholesky factorization proves to be quite fruitful in the study of non standard orthogonality such as multiple, matrix, Sobolev, and multivariate orthogonality as well as orthogonality on the unit circle of the complex plane, and have successfully found its way into applications in random matrices, Toda lattices, integrable systems, Riemann-Hilbert problems, Painlevé equations, and Darboux transformations, among others topics.…”