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.