$q$-poly-Bernoulli numbers and $q$-poly-Cauchy numbers with a parameter by Jackson's integrals

TL;DR

This paper introduces generalized $q$-poly-Bernoulli and poly-Cauchy polynomials with a parameter using Jackson's integrals, extending classical numbers and exploring their properties and relations with Stirling numbers.

Contribution

It defines new $q$-poly-Bernoulli and poly-Cauchy polynomials with a parameter, generalizing known numbers and polynomials, and investigates their properties and interrelations.

Findings

Defined $q$-poly-Bernoulli and poly-Cauchy polynomials with a parameter

Established connections with Stirling numbers and weighted Stirling numbers

Derived relations between generalized poly-Bernoulli and poly-Cauchy polynomials

Abstract

We define $q$-poly-Bernoulli polynomials $B_{n,\rho,q}^{(k)}(z)$ with a parameter $\rho$, $q$-poly-Cauchy polynomials of the first kind $c_{n,\rho,q}^{(k)}(z)$ and of the second kind $\widehat c_{n,\rho,q}^{(k)}(z)$ with a parameter $\rho$ by Jackson's integrals, which generalize the previously known numbers and polynomials, including poly-Bernoulli numbers $B_n^{(k)}$ and the poly-Cauchy numbers of the first kind $c_n^{(k)}$ and of the second kind $\widehat c_n^{(k)}$. We investigate their properties connected with usual Stirling numbers and weighted Stirling numbers. We also give the relations between generalized poly-Bernoulli polynomials and two kinds of generalized poly-Cauchy polynomials.