Hankel Determinants of sequences related to Bernoulli and Euler Polynomials

Abstract

We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and 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.…”

Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

“…We begin this section with some necessary background on the connection between orthogonal polynomials and Hankel determinants. All this is well-known and can also be found in concise form in [4] and [5]. We repeat this material here for easy reference, and to make this paper self-contained.…”

“…In the recent paper [4] we used the connection with orthogonal polynomials and continued fractions to find new evaluations of Hankel determinants of certain subsequences of Bernoulli and Euler polynomials. This was followed in [5] by evaluations of Hankel determinants of various other sequences related to Bernoulli and Euler numbers and polynomials. B n (x) t n n!…”

“…The paper [5] also contains a list of all "Hankel-Bernoulli/Euler" identities known to us. It turned out that the majority of such identities, when written in a standard way, are of a very specific form.…”