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 A $q$-analogue of $w$-Bernoulli numbers and their applications Bull. Korean Math. Soc. 2001 Vol. 38, No. 2, 399-412 Jin-Woo Son and Douk Soo Jang Kyungnam University, Kyungnam University Abstract : 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 Downloads: Full-text PDF