Christian Krattenthaler

A determinant identity for moments of orthogonal polynomials that implies Uvarov's formula for the orthogonal polynomials of rationally related densities

(20 pages)

Abstract. Let pn(x), n=0,1,..., be the orthogonal polynomials with respect to a given density dμ(x). Furthermore, let dν(x) be a density which arises from dμ(x) by multiplication by a rational function in x. We prove a formula that expresses the Hankel determinants of moments of dν(x) in terms of a determinant involving the orthogonal polynomials pn(x) and associated functions qn(x)=∫ pn(u) dμ(u)/(x-u). Uvarov's formula for the orthogonal polynomials with respect to dν(x) is a corollary of our theorem. Our result generalises a Hankel determinant formula for the case where the rational function is a polynomial that existed somehow hidden in the folklore of the theory of orthogonal polynomials but has been stated explicitly only relatively recently (see [arXiv:2101.04225]). Our theorem can be interpreted in a two-fold way: analytically or in the sense of formal series. We apply our theorem to derive several curious Hankel determinant evaluations.

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