Certain identities, connection and explicit formulas for the Bernoulli, Euler numbers and Riemann zeta -values

TL;DR

This paper derives new identities, recurrence relations, and explicit formulas for Bernoulli and Euler numbers and Riemann zeta values using properties of Sheffer sequences and the Kontorovich-Lebedev transform.

Contribution

It introduces novel identities and formulas connecting Bernoulli, Euler numbers, and zeta values through Sheffer sequences and integral transforms.

Findings

New identities for Bernoulli and Euler numbers

Recurrence relations for zeta function values

Explicit formulas involving Sheffer sequences

Abstract

Various new identities, recurrence relations, integral representations, connection and explicit formulas are established for the Bernoulli, Euler numbers and the values of Riemann's zeta function. To do this, we explore properties of some Sheffer's sequences of polynomials related to the Kontorovich-Lebedev transform.