In the paper, we introduce an analogue of Haar distribution based on
c3bcebefd4a27f02e9d" title="Click to view the MathML source">(ρ,q)-numbers, as follows:
By means of this distribution, we derive
c3bcebefd4a27f02e9d" title="Click to view the MathML source">(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
c3bcebefd4a27f02e9d" title="Click to view the MathML source">(ρ,q)-numbers. Finally, we construct
c3bcebefd4a27f02e9d" title="Click to view the MathML source">(ρ,q)-Bernoulli numbers and polynomials derived from
c3bcebefd4a27f02e9d" title="Click to view the MathML source">(ρ,q)-Volkenborn integral and obtain some their properties.