[Paper Review] Hankel Determinants of sequences related to Bernoulli and Euler Polynomials

Karl Dilcher, Lin Jiu|arXiv (Cornell University)|Jul 20, 2020

Advanced Mathematical Identities14 references4 citations

TL;DR

This paper evaluates Hankel determinants of sequences related to Bernoulli and Euler polynomials using connections to orthogonal polynomials and derivatives of these polynomials. It derives new closed-form evaluations, including $ H_{2m+1}(kE_{k-1}) = (-1)^{m+1}2^{4m(m+1)}\prod_{\ell=1}^{m}\ell!^8 $, and organizes known and novel identities for sequences involving Bernoulli and Euler numbers and their special values.

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.

Motivation & Objective

• To derive new closed-form evaluations of Hankel determinants for sequences related to Bernoulli and Euler polynomials.
• To extend known results by introducing a method based on derivatives of Bernoulli and Euler polynomials.
• To unify and organize existing and novel Hankel determinant identities involving Bernoulli and Euler numbers and their special values.
• To explore connections between Hankel determinants, orthogonal polynomials, and continued fractions in the context of special number sequences.
• To provide systematic evaluations for generalized sequences, including those related to Dirichlet characters and higher-order Euler numbers.

Proposed method

• Utilizes generating functions for Bernoulli and Euler polynomials as foundational tools for sequence construction.
• Applies the derivative-based method to transform Hankel determinant problems into manageable algebraic forms.
• Employs connections with classical orthogonal polynomials and continued fractions to evaluate Hankel determinants.
• Uses symbolic notation, such as $ \mathcal{B}^j $, to represent Bernoulli numbers in generalized identities.
• Applies known results from prior works (e.g., [6], [8], [9]) to derive new identities through algebraic manipulation and substitution.
• Applies reflection and symmetry properties of Bernoulli and Euler polynomials to simplify determinant expressions.

Experimental results

Research questions

• RQ1 What are the Hankel determinant evaluations for sequences formed by sums and differences of Bernoulli polynomials with the same index?
• RQ2 How can the derivatives of Bernoulli and Euler polynomials be used to derive new Hankel determinant identities?
• RQ3 What are the Hankel determinant evaluations for generalized Bernoulli polynomials associated with Dirichlet characters of small conductor?
• RQ4 How do Hankel determinants of sequences like $ kE_{k-1} $ behave, and what closed-form expressions can be derived?
• RQ5 What is the relationship between Hankel determinants of Euler numbers and those of the combinatorially defined sequence $ \mathbf{E}_n $, and how can identities be transferred between them?

Key findings

• The Hankel determinant $ H_n(E_k) $ for Euler numbers is given by $ (-1)^{\binom{n+1}{2}}\prod_{\ell=1}^{n}\ell!^2 $, a known result confirmed and contextualized.
• For the sequence $ kE_{k-1} $, the odd-order Hankel determinant is $ H_{2m+1}(kE_{k-1}) = (-1)^{m+1}2^{4m(m+1)}\prod_{\ell=1}^{m}\ell!^8 $, with even-order determinants vanishing.
• A general identity is derived for Hankel determinants of the form $ H_n(\mathcal{B}^{k+2}({\mathcal{B}}+1)_{a-1}({\mathcal{B}}+1)_{b-1}(-{\mathcal{B}}+1)_{c-1}(-{\mathcal{B}}+1)_{d-1}) $, involving factorials and powers.
• The method successfully evaluates Hankel determinants for generalized Bernoulli polynomials under Dirichlet characters with small conductors.
• The paper establishes a bridge between combinatorial sequences $ \mathbf{E}_n $ and classical Euler numbers, enabling transfer of Hankel determinant identities.
• The study confirms and extends results from Han [9] and Fulmek and Krattenthaler [8], particularly for mixed sequences involving $ \mathbf{E}_k $ and $ \mathbf{E}_k/(k)! $.