$q$-Extensions of Some Results Involving the Luo-Srivastava Generalizations

Abstract

Carlitz firstly defined the $q$-Bernoulli and $q$-Euler polynomials [\textit{Duke Math. J.}, \textbf{15} (1948), 987--1000]. Recently, M. Cenkci and M. Can [\textit{Adv. Stud. Contemp. Math.}, \textbf{12} (2006), 213--223], J. Choi, P. J. Anderson and H. M. Srivastava [ \textit{Appl. Math. Comput.}, {\bf199} (2008), 723--737] further defined the $q$-Apostol-Bernoulli and $q$-Apostol-Euler polynomials. In this paper, we show the generating functions and basic properties of the $q$-Apostol-Bernoulli and $q$-Apostol-Euler polynomials, and obtain some relationships between the $q$-Apostol-Bernoulli and $q$-Apostol-Euler polynomials which are the corresponding $q$-extensions of some known results. Some formulas in series of $q$-Stirling numbers of the second kind are also considered.