Congruences concerning Bernoulli numbers and Bernoulli polynomials

Let {B-n(x)} denote Bernoulli polynomials. In this paper we generalize Kummer's congruences by determining Bk(p-1)+b(x)/(k(p - 1) + b)(mod p(n), where p is an odd prime, x is a p-integral rational number and p -1 + b. As applications we obtain explicit formulae for Sigma(x=1)(p-1) (1/x(k)) mod p(3)), Sigma(x=1)((p-1)/2) (1/x(k))(mod p(3)), (p - 1)!(mod p(3)) and A(r)(m, p)(modp), where k is an element of {1,2,..., p - 1} and A(r)(m, p) is the least positive solution of the congruence px = r(mod m). We also establish similar congruences for generalized Bernoulli numbers {B-n,B-chi}. (C) 2000 Elsevier Science B.V. All rights reserved.