On the q-Extensions of the Bernoulli and Euler Numbers, Related Identities and Lerch Zeta Function

Abstract

Recently, $\lambda$-Bernoulli and $\lambda$-Euler numbers are studied in [5, 10]. The purpose of this paper is to present a systematic study of some families of the $q$-extensions of the $\lambda$-Bernoulli and the $\lambda$-Euler numbers by using the bosonic $p$-adic $q$-integral and the fermionic $p$-adic $q$-integral. The investigation of these $\lambda$-$q$-Bernoulli and $\lambda$-$q$-Euler numbers leads to interesting identities related to these objects. The results of the present paper cover earlier results concerning $q$-Bernoulli and $q$-Euler numbers. By using derivative operator to the generating functions of $\lambda$-$q$-Bernoulli and $\lambda$-$q$-Euler numbers, we give the $q$-extensions of Lerch zeta function.