(ρ,q)-Volkenborn integration

In the paper, we introduce an analogue of Haar distribution based on (ρ,q)-numbers, as follows:μρ,q(a+pNZp)=ρpN[pN]ρ,q(qρ)a. By means of this distribution, we derive (ρ,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 (ρ,q)-numbers. Finally, we construct (ρ,q)-Bernoulli numbers and polynomials derived from (ρ,q)-Volkenborn integral and obtain some their properties.