We can define p-adic Bernoulli polynomials by using q-integral on $\mathbb{Z}\_p$ and Taekyun Kim's method.


But how can we define p-adic poly-Bernoulli numbers and polynomials by using integral on $\mathbb{Z}\_p$? 



p-adic poly-Bernoulli numbers
I think what you are looking for is:


$$B\_n^{k}(x)=\frac{Li\_k (1-e^{-t})}{t}\int\_{\mathbb{Z}\_p}(x+y)^ndy$$


for $n\geq0$, and $Li\_k$ the polylogarithmic function.


See for example, "[Poly-Bernoulli Polynomials and Their Applications](http://www.m-hikari.com/ijma/ijma-2014/ijma-29-32-2014/kimIJMA29-32-2014-2.pdf)", by Kim, Jang and Seo.


