Result: SOME REMARKS ON A q-ANALOGUE OF BERNOULLI NUMBERS: Some remarks on a \(q\)-analogue of Bernoulli numbers

Let \({\mathbb Z} _p\) be the ring of \(p\)-adic integers, \({\mathbb Q} _p\) the field of \(p\)-adic numbers, and \({\mathbb C} _p\) the completion of the algebraic closure of \({\mathbb Q} _p\). Assume also that \(q\) is an element of \({\mathbb C} _p\) satisfying the inequality \(|q - 1|< p ^{-1/(p-1)}\), where \(|. |\) is the \(p\)-adic valuation of \({\mathbb C} _p\) determined by the equality \(|p|= p ^{-1}\). Using the \(p\)-adic \(q\)-integral introduced by \textit{T. Kim} [J. Number Theory 76, 320-329 (1999; Zbl 0941.11048)], the authors define, for each integer \(n \geq 0\), a \(p\)-adic \(q\)-Bernoulli number \(B _n ^* (q)\) and a \(q\)-Bernoulli polynomial \(B _n ^* (x; q)\) as a \(p\)-adic \(q\)-analogue to the ordinary Bernoulli number and Bernoulli polynomial, respectively. They study the sequences \(B _n ^* (q): n \geq 0\) and \(B _n ^* (x; q): n \geq 0\) in various aspects, e.g. find identities relating their elements, prove that \(\lim _{q \to 1} B _n ^* (x; q) = B _n (x)\) and \(\lim_{q \to 1} B _n ^* (q) = B _n\), where \(B _n (x)\) and \(B _n\) are the ordinary \(n\)-th Bernoulli polynomial and \(n\)-th Bernoulli number, respectively. The obtained results are used for computing some \(p\)-adic \(q\)-integrals and for constructing \(p\)-adic distributions. Finally, the authors construct a modification of \textit{H. Tsumura}'s function [see Tokyo J. Math. 10, 281-293 (1987; Zbl 0641.12007)] and determine its values at non-positive integers.