An explicit Formula for Bernoulli Polynomials with a $q$ Parameter in Terms of $r$-Whitney Numbers

We define the Bernoulli polynomials with a $q$ parameter in terms of $r$-Whitney numbers of the second kind. Some algebraic properties and combinatorial identities of these polynomials are given. Also, we obtain several relations between the Cauchy and Bernoulli polynomials with a $q$ parameter in terms of $r$-Whitney numbers of both kinds.