In an earlier paper [Rep. Fac. Sci. Eng., Saga Univ., Math. 23, 1-7 (1995; Zbl 0820.11071)] the authors introduced a -adic -integral giving an integral representation of the -Bernoulli numbers . Now this approach is used for defining -Bernoulli polynomials. Then the -adic -Bernoulli measures are constructed and used to define -adic --series, which interpolate -Bernoulli numbers.
Note that there exists another sequence of -Bernoulli numbers [introduced by L. Carlitz, Duke Math. J. 15, 987-1000 (1948; Zbl 0032.00304)]. The corresponding -adic --functions were found by N. Koblitz [J. Number Theory 14, 332-339 (1982; Zbl 0501.12020)].