Analogues of Faulhaber's Formula for Poly-Bernoulli and Type 2 Poly-Bernoulli Polynomials

Faulhaber's formula expresses sums of powers of consecutive integers in terms of Bernoulli polynomials. Here we would like to find analogous ones to the Faulhaber's formula for poly-Bernoulli and type 2 poly-Bernoulli polynomials.