Degenerate and n-adic versions of Kummer's congruences for values of Bernoulli polynomials

Abstract

We first prove an analogue of Kummer's congruences for expressions involving the degenerate Bernoulli polynomials which were introduced by Carlitz, by relating them to the general theory of “degenerate number sequences” developed in a recent article. These congruences extend, for example, known congruences for the Genocchi numbers. We also give versions of Kummer's congruences modulo powers of a general positive integer n for Bernoulli polynomials with n-adic integer argument, and similar congruences for generalized Bernoulli polynomials, which extend recent results of Z.-W. Sun and of the author.