Degenerate Hurwitz-Lerch Type Families of Poly-Bernoulli and Poly-Cauchy Numbers with Parameters $(α,a)

Abstract

In this paper, we introduce degenerate Hurwitz-Lerch type families of poly-Bernoulli and poly-Cauchy numbers with parameters $(α,a)$. These families are defined by means of degenerate Hurwitz-Lerch type zeta functions and their factorial analogues. We establish the generating functions of these numbers and derive explicit formulas in terms of the degenerate Stirling numbers of the first and second kinds. In particular, we obtain closed-form expressions for the degenerate two-parameter Hurwitz-Lerch type poly-Bernoulli numbers and for the corresponding poly-Cauchy numbers of the first and second kinds. These results extend several known identities for classical and generalized poly-Bernoulli and poly-Cauchy numbers.