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

Mustafa Alkan, Yilmaz Simsek

By using periodic functions 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 [1]. Then using this generating function, we a construct new $L$-type series. By using periodic functions, we derive decomposition of the generating functions for the $q$-Euler numbers and polynomials. Applying the Mellin transformation to the decomposition of the generating functions, we 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 polynomials and numbers.