Hurwitz--Lerch Type Families of Poly-Bernoulli and Poly-Cauchy Numbers and Polynomials with Parameters $(\alpha,a)$

TL;DR

This paper introduces new parameterized families of poly-Bernoulli and poly-Cauchy numbers and polynomials based on the Hurwitz-Lerch zeta function, expanding classical sequences and uncovering new mathematical structures.

Contribution

It extends classical poly-Bernoulli and poly-Cauchy sequences by incorporating parameters and the Hurwitz-Lerch zeta function, providing new formulas and connections.

Findings

Derived generating functions and explicit formulas

Established recurrences and structural properties

Connected new sequences with zeta functions and polylogarithms

Abstract

This study introduces $(\alpha,a)$-parameterized Hurwitz-Lerch type poly-Bernoulli and poly-Cauchy numbers and polynomials, extending classical sequences through the Hurwitz-Lerch zeta function. We derive generating functions, recurrences, and explicit formulas, revealing deeper structures and connections with zeta functions and polylogarithms. The results enrich existing theory and open new directions in analytic number theory and combinatorics.