A \(q\)-analogue of \(w\)-Bernoulli numbers and their applications (Q2732168)

The \(q\)-analogues of the \(w\)-Bernoulli numbers are defined by the relations NEWLINE\[NEWLINE B_0(w;q)=1,\quad w(qB(w;q)+1)^n-B_n(w;q)=\begin{cases} 1,&\text{for \(n=1\),}\\ 0,&\text{for \(n=1\)},\end{cases} NEWLINE\]NEWLINE where \(B^i(w;q)\) is replaced by \(B_i(w;q)\). Here \(w,q\in \mathbb C\), \(| w| <1\), \(| q| <1\). The authors find an identity for the generating function of the sequence \(\{B_n(w;q)\}\) which implies an identity for a sum of products of \(B_n(w;q)\).NEWLINENEWLINEIn the second part of the paper it is assumed that \(q\) and \(w\) belong to \(\mathbb C_p\), with \(| 1-q| _p<p^{\frac{1}{p-1}}\), \(| 1-w^{- 1}| _p\geq 1\). A \(p\)-adic Stirling type formula involving \(B_n(w;q)\) is obtained.