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

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.