Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

Abstract

Hankel determinants of sequences related to Bernoulli and Euler numbers have been studied before, and numerous identities are known. However, when a sequence is shifted by one unit, the situation often changes significantly. In this paper we use classical orthogonal polynomials and related methods to prove a general result concerning Hankel determinants for shifted sequences. We then apply this result to obtain new Hankel determinant evaluations for a total of 13 sequences related to Bernoulli and Euler numbers, one of which concerns Euler polynomials.

Hankel Determinants of Certain Sequences Of Bernoulli Polynomials: A Direct Proof of an Inverse Matrix Entry from Statistics

“…It is the purpose of this paper, later in Section 5, to give a direct computational proof, based on the corresponding Hankel determinant of certain Bernoulli polynomials B n (x). These sequences are different from the recent work [7,8,9], by Dilcher and the first author, on the Hankel determinants of sequences related to Bernoulli and Euler polynomial.…”

“…All the necessary background stated here in this section can be found in [7,8,9], in concise form. We repeat this material here for easy reference, and to make this paper self-contained.…”