Some New Families of Generalized Euler and Genocchi Polynomials

The main object of this paper is to introduce and investigate a new generalization of the family of Euler polynomials by means of a suitable generating function. We establish several interesting properties of these general polynomials and derive explicit representations for them in terms of a certain generalized Hurwitz-Lerch Zeta function and in series involving the familiar Gaussian hypergeometric function. Finally, we propose an analogous generalization of the closely-related Genocchi polynomials and show how we can fruifully exploit some potentially useful linear connections of all these three important families of generalized Bernoulli, Euler and Genocchi polynomials with one another.