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

TL;DR

This paper derives a new formula linking Hankel determinants of linear combinations of moments of orthogonal polynomials to a smaller determinant of the polynomials themselves, with multiple proofs and applications to classical sequences.

Contribution

It introduces a novel, explicitly proven formula connecting Hankel determinants of combined moments to orthogonal polynomials, expanding understanding and computational methods.

Findings

Provides four different proofs of the formula.

Enables computation of Hankel determinants in singular cases.

Shows linear recurrence relations for classical combinatorial sequences.

Abstract

We present a formula that expresses the Hankel determinants of a linear combination of length $d+1$ of moments of orthogonal polynomials in terms of a $d\times d$ 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} (2015), 1253-1274] (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Br\'ezin and Hikami [Comm. Math. Phys. 214 (2000), 111-135], 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, Schr\"oder numbers, Riordan numbers, and Fine numbers.