The Families of L-Series Associated with Decomposition of the Generating Functions

Abstract

By using a periodic function from the nonnegative integers to the complex numbers, we generalize the generating function of the q-Apostol type Eulerian polynomials and numbers attached the character defined in <cite>AL-SIM</cite>. Then using the generating function of the q-Eulerian polynomials attached a periodic function χ, we construct new L-type series. By using a periodic functions, we derive decomposition of the generating functions for the q-Euler numbers and polynomials. Also applying the Mellin transformation to the decomposition of the generating functions, we also introduce and investigate the various properties of a certain new family of the Dirichlet type l-series and the Dirichlet l-function. Finally, we derive many potentially useful results involving these functions and numbers.