Hankel determinants of linear combinations of moments of orthogonal polynomials, II

Abstract

We present a formula that expresses the Hankel determinants of a linear combination of length $$d+1$$ d + 1 of moments of orthogonal polynomials in terms of a $$d\times d$$ d × d determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi (J Math Anal Appl 431:1253–1274, 2015) (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami (Commun Math Phys 214:111–135, 2000), and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.

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

“…(6) In [5], three proofs of the special case of Theorem 1 where k = 0 -that is, of Corollary 2 -are given, one using the method of condensation, one using classical results from the theory of orthogonal polynomials, and one using a vanishing argument. It is interesting to note that neither the second nor the third proof seem to extend to a proof of Theorem 1, only the condensation argument does.…”

“…8.4.1] (although incorrectly) and by Elouafi [2, Theorem 1] (however with an incomplete proof). Three fundamentally different proofs are given in [5]: one due to this author, one following Lascoux's arguments, and one completing Elouafi's arguments.The purpose of this article is to present and prove a generalisation of the aforementioned formula that is inspired by Uvarov's formula [8,9] for the orthogonal polynomials with…”

“…8.4.1] (although incorrectly) and by Elouafi [2, Theorem 1] (however with an incomplete proof). Three fundamentally different proofs are given in [5]: one due to this author, one following Lascoux's arguments, and one completing Elouafi's arguments.…”

“…Introduction. Recently,in [5] this author discovered a formula that expresses the Hankel determinant of linear combinations of moments of orthogonal polynomials in terms of a determinant involving these orthogonal polynomials. A literature search revealed that this formula existed in a hidden form behind a theorem (cf.…”

“…(3) The main theorem in [5] is equivalent to the special case of Theorem 1 where k = 0. It is worth stating this special case separately.…”