Isospectral Matrices and Properties of Special Polynomials

by Oksana Bihun (University of Colorado at Colorado Springs, CO, USA)

Monday 5 Jun 2017, 14:30 → 16:00 Europe/Rome

Aula Conversi (Dipartimento di Fisica - Ed. G.Marconi)

We will present a new matrix M, and several other new matrices, with the following properties. The matrix M is defined in terms of the zeros and the parameters of a N-th degree polynomial p_N. The polynomial p_N belongs to an orthogonal polynomial family from the top of the q-Askey scheme. The matrix M is isospectral, that is, its neat eigenvalues do not depend on the zeros of the polynomial p_N and depend only on some of its parameters. This remarkable matrix encodes multiple new algebraic relations satisfied by the zeros of the polynomial p_N. The matrix M is discovered via a linearization of the system of ODEs satisfied by the zeros of a certain polynomial, with time-dependent coefficients, which itself solves a certain Differential q-Difference Equation (DqDE). This is a joint work with Francesco Calogero.