Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers

TL;DR

This paper generalizes Bernoulli and zeta identities, explores their relations through determinants and Mellin transforms, and introduces new representations and asymptotics for Hurwitz numbers, connecting various special functions and number theory concepts.

Contribution

It extends classical Bernoulli and zeta identities to Hurwitz and Lerch functions, providing new recurrences, determinant relations, and integral representations for Hurwitz numbers.

Findings

Generalized Bernoulli and zeta identities and recurrences.

New integral and series representations for Hurwitz numbers.

Asymptotic formulas for Hurwitz numbers derived.

Abstract

The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet $L$ series, to Eisenstein series, and to general Mellin transforms. The Hurwitz numbers $\tilde{H}_n$ occur in the Laurent expansion about the origin of a certain Weierstrass $\wp$ function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general $\wp$ functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence.