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 dtimes 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.
Highlights
1 Introduction The purpose of this article is to put to the fore a fundamental formula for orthogonal polynomials that is implicitly hidden in the classical literature on orthogonal polynomials
My literature search led me to discover that the formula is stated in Lascoux’s book [11], albeit incorrectly, but with a correct proof
We point out that the above treatment of the “Motzkin case” is one which only applies in a specific situation, whereas the above treatment of the “Schröder case” works for all moment sequences for orthogonal polynomials which are generated by a three-term recurrence (1.2) where the coefficient sequencesi≥0 andi≥0 become constant eventually, that is, where si ≡ s and ti ≡ t for large enough i
Summary
The purpose of this article is to put to the fore a fundamental formula for orthogonal polynomials that is implicitly hidden in the classical literature on orthogonal polynomials. In reaction to [10], Arno Kuijlaars pointed out to me that the formula appears in [1], in which a result due to Brézin and Hikami [2] is cited Both papers contain correct statement and (different) proofs, they use random matrix language. I consulted Mourad Ismail and asked him if he knows the formula, respectively can refer me to a source in the literature He immediately pointed out that the right-hand side determinant of (1.1) features in “Christoffel’s theorem" about the orthogonal polynomials with respect to the measure defined by the density id=−11(x + xi ) dμ(x) (with dμ(x) the density of the original orthogonality measure), that is, in [14, Theorem 2.5] respectively [7, Theorem 2.7.1]. Result [2, Eq (14)] into the language that we use here to see that it is equivalent to (1.1)
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