... prime p, ℤp, ℚp, and ℂpdenote the ring of p -adic integers, the field of p-adic numbers, and the completion of the algebraic closure of ℚp, respectively. Let | · |pbe the p-adic norm on ... 10 of 16Remark 3.2. The remarkable point is that the series on the right-hand side of Lemma3.1 is uniformly convergent in the wider sense.From (2.13 )and (2.14), we define the q-Bernoulli and ... contributions The authors have equal contributions to each part of this paper. All the authors read and approved the finalmanuscript.Competing interests The authors declare that they have no...