Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

TL;DR

This paper investigates Hankel determinants of shifted Bernoulli and Euler number sequences, using orthogonal polynomial methods to derive new determinant evaluations and general results for these shifted sequences.

Contribution

The paper introduces a general theorem for Hankel determinants of shifted sequences and applies it to evaluate determinants for 13 Bernoulli and Euler related sequences.

Findings

New Hankel determinant evaluations for 13 sequences

A general result for shifted sequence determinants

Application of orthogonal polynomials to sequence analysis

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.