On the families of q-Euler polynomials and their applications

In this paper, we focus on applications of q-Euler polynomials and obtain some new combinatorial relations by using p-adic q  -integral on ZpZp. Moreover, we derive distribution formula (Multiplication Theorem) for Dirichlet type of q  -Euler numbers and polynomials with weight αα. Also we apply the method of analytic continuation of q-Euler polynomials which is the main result of this paper.