Some remarks on a q-analogue of Bernoulli numbers

Abstract

Using the $p$-adic $q$-integral due to T. Kim [4], we define a number $B_n^*(q)$ and a polynomial $B_n^*(x;q)$ which are $p$-adic $q$-analogue of the ordinary Bernoulli number and Bernoulli polynomial, respectively. We investigate some properties of these. Also, we give slightly different construction of Tsumura's $p$-adic function $\ell_{p}(u,s,\chi)$ [14] using the $p$-adic $q$-integral in [4].