LAGUERRE-ANGELESCO MULTIPLE ORTHOGONAL POLYNOMIALS ON AN R-STAR

by MARJOLEIN LEURS (KU LEUVEN)

Location: 200B.02.18, Leuven
Time: Thursday October 25, 2018 at 14:00

Among the classic orthogonal polynomials, the Laguerre polynomials can be obtained from the Jacobi polynomials by letting one parameter tend to infinity. Now we generalize this to multiple orthogonal polynomials with respect to measures supported on an $r$-star. In a previous talk I introduced the Jacobi-Angelesco polynomials on an $r$-star which are orthogonal with respect to a Jacobi weight $

x

^{\beta}(1-x^r)^{\alpha}$. The Laguerre-Angelesco polynomials are orthogonal with respect to a Laguerre weight function $

x

^{\beta}e^{-x^r}$. Again we obtain the Laguerre-Angelesco polynomials by letting one parameter tend to infinity in the Jacobi-Angelesco polynomials.

In this talk I present recent results of the Laguerre-Angelesco multiple orthogonal polynomials on an $r$-star. There are two types of multiple orthogonal polynomials. For the type I Laguerre-Angelesco polynomials we give explicit expressions for the polynomials on and near the diagonal, a differential equation and the asymptotic behavior of the zeros of the polynomials. Note that we first need to rescale our polynomials in order that the zeros lie in a compact interval. The type II Laguerre-Angelesco polynomials were already studied. We state the Rodrigues formula and a differential equation. For each type we also show how these polynomials are obtained from the Jacobi-Angelesco polynomials.