In the paper, we in
troduce an analogue of Haar dis
tribution based on
trieve&_eid=1-s2.0-S0022314X16302025&_mathId=si3.gif&_user=111111111&_pii=S0022314X16302025&_rdoc=1&_issn=0022314X&md5=881a5d65f053dc3bcebefd4a27f02e9d" title="Click to view the MathML source">(ρ,q)-numbers, as follows:
By means of this dis
tribution, we derive
trieve&_eid=1-s2.0-S0022314X16302025&_mathId=si3.gif&_user=111111111&_pii=S0022314X16302025&_rdoc=1&_issn=0022314X&md5=881a5d65f053dc3bcebefd4a27f02e9d" 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
trieve&_eid=1-s2.0-S0022314X16302025&_mathId=si3.gif&_user=111111111&_pii=S0022314X16302025&_rdoc=1&_issn=0022314X&md5=881a5d65f053dc3bcebefd4a27f02e9d" title="Click to view the MathML source">(ρ,q)-numbers. Finally, we cons
truct
trieve&_eid=1-s2.0-S0022314X16302025&_mathId=si3.gif&_user=111111111&_pii=S0022314X16302025&_rdoc=1&_issn=0022314X&md5=881a5d65f053dc3bcebefd4a27f02e9d" title="Click to view the MathML source">(ρ,q)-Bernoulli numbers and polynomials derived from
trieve&_eid=1-s2.0-S0022314X16302025&_mathId=si3.gif&_user=111111111&_pii=S0022314X16302025&_rdoc=1&_issn=0022314X&md5=881a5d65f053dc3bcebefd4a27f02e9d" title="Click to view the MathML source">(ρ,q)-Volkenborn integral and obtain some their properties.