On \(p\)-adic \(q\)-\(L\)-functions and sums of powers
Let \(q\in \mathbb C_p\), \(|q-1|_p<p^{-1/(p-1)}\). The author finds an explicit expression for the sum of \(\frac{q^j}{[j]^r}\) taken over natural numbers \(j\), such that \(1\leq j\leq np\) and \(j\) is prime to \(p\), via a \(q\)-analog of the \(p\)-adic \(L\)-functions. Here \([j]=\frac{1-q^j}{1-q}\). As special cases and corollaries, this result includes earlier ones by Andrews, Washington, Barsky, and others. A \(q\)-analog, for \(q\in \mathbb C\), \(|q|<1\), is also found for a recurrence for Bernoulli numbers obtained by \textit{E. Y. Deeba} and \textit{D. M. Rodriguez} [Am. Math. Mon. 98, 423-426 (1991; Zbl 0743.11012)].
scientific article; zbMATH DE number 1792423