Translation-invariant \(p\)-adic integral on \(\mathbb{Z}_p\) (Q2767903)

A \(q\)-analogue of the Volkenborn \(p\)-adic integral (introduced by \textit{T. Kim} [Arch. Math. 76, 190-195 (2001; Zbl 0986.11010)], and \textit{T. Kim, D.-W. Park} and \textit{S.-H. Rim} [J. Phys. A, Math. Gen. 34, 7633-7638 (2001; Zbl 1002.11088)]) gives rise to the \(q\)-Bernoulli numbers NEWLINE\[NEWLINE B_{n,q}=\int_{\mathbb Z_p}x^n d\mu_q(x). NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEThe authors find generating functions for \(B_{n,q}\) and certain more general sequences, and study relations between \(B_{n,q}\) and other generalizations of the Bernoulli numbers [\textit{T. Kim}, Rep. Fac. Sci. Eng., Saga Univ., Math. 22, 21-26 (1994; Zbl 0802.11007) and \textit{K. Shiratani} and \textit{S. Yamamoto}, Mem. Fac. Sci., Kyushu Univ., Ser. A 39, 113-125 (1985; Zbl 0574.12017)].NEWLINENEWLINEFor the entire collection see [Zbl 0972.00006].