A $q$-analogue of $w$-Bernoulli numbers and their applications

摘要

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 $eta_i (q)$ 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.