A unified approach for the Hankel determinants of classical combinatorial numbers

Abstract

Please cite this article in press as: M. Elouafi, A unified approach for the Hankel determinants of classical combinatorial numbers, J. Math. Anal. Appl. (2015), http://dx.Abstract. We give a general formula for the determinants of a class of Hankel matrices which arise in combinatorics theory. We revisit and extend existant results on Hankel determinants involving the sum of consecutive Catalan, Motzkin and Schroder numbers and we prove a conjecture in [20] about the recurrence relations satisfied by the Hankel transform of linear combinations of Catalans numbers.

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

“…3. Later I found the earlier mentioned occurrences of the formula in [1,6,11] and [2]. Lascoux's argument (the one in [1] is essentially the same), which follows the classical literature of orthogonal polynomials (but is presented in [11] in his very personal language), is presented in Sect.…”

“…The point of (1.1) in this context is that it provides a compact formula for n × n Hankel determinants of a fixed linear combination of d + 1 successive elements of a (moment) sequence (the lefthand side of (1.1)) that does not "grow" with n. (The right-hand side is a "fixed-size" formula for fixed d; the dependence on n is in the index of the orthogonal polynomials.) Elouafi provides numerous applications of Theorem 1 to the evaluation of Hankel determinants of linear combinations of Catalan, Motzkin, and Schröder numbers in [6,Sect. 3].…”

“…In order to apply the proposition, we must evaluate derivatives of the orthogonal polynomials at −x j . How to accomplish this for the orthogonal polynomials corresponding to Motzkin and Schröder numbers as moments by a recursive approach, is described in [6,Sects. 3.2 and 3.3].…”

“…My literature search led me to discover that the formula is stated in Lascoux's book [11], albeit incorrectly, but with a correct proof. Subsequently, I realised that the formula is stated correctly by Elouafi in [6], however with an incomplete proof. Finally, 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.…”

“…Lascoux refers to "the Bazin formula" without any reference. He may be excused for that: a glance at the index of[11] leads one to[11, Lemma A.1.1] 6. The proof of Lemma 4 in[6] only works for pairwise distinct α i 's.…”