Bulletin of the
Korean Mathematical Society BKMS

In this paper, we consider that the $q$-analogue of $w$-Bernoulli numbers $\Cal B_i (w,q)$. And we calculate the sums of products of two $q$-analogue of $w$-Bernoulli numbers $\Cal B_i (w,q)$ in complex cases. From this result, we obtain the Euler type formulas of the Carlitz's $q$-Bernoulli numbers $\beta_i (q)$ and $q$-Bernoulli numbers $\Cal B_i(q)$. And we also calculate the $p$-adic Stirling type series by the definition of $\Cal B_i(w,q)$ in $p$-adic cases.

Keywords: Bernoulli numbers, $p$-adic Stirling series, Euler formula

MSC numbers: 11B68, 11E95