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

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 ${\hat{c}}_{n,\rho ,q}^{(k)}(z)$ 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 ${\hat{c}}_n^{(k)}$. 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.

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• Formulas involving Cauchy polynomials, Bernoulli polynomials, and generalized Stirling numbers