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 $\mathbb{Z}_p$?

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$$B_n^{k}(x)=\frac{Li_k (1-e^{-t})}{t}\int_{\mathbb{Z}_p}(x+y)^ndy$$
for $n\geq0$, and $Li_k$ the polylogarithmic function.
Are those $t$'s supposed to be $n$'s? –  Hurkyl Sep 30 at 13:18