(?,q)-Volkenborn integration

Abstract

In the paper, we introduce an analogue of Haar distribution based on (rho, q)-numbers, as follows:

mu(rho,q) (a + p(N)Z(p)) = rho(pN)/[p(N)](rho,q) (q/rho)(a)

By means of this distribution, we derive (rho, q)-analogue of Volkenborn integration which is a new generalization of Kim's q-Volkenborn integration defined in [11]. From this definition, we investigate some properties of Volkenborn integration based on (rho, q)-numbers. Finally, we construct (rho, q)-Bernoulli numbers and polynomials derived from (rho, q)-Volkenborn integral and obtain some their properties. (C) 2016 Elsevier Inc. All rights reserved.