Congruences involving Bernoulli polynomials

Let {Bn(x)} be the Bernoulli polynomials. In the paper we establish some congruences for Bj(x) (mod p n), where p is an odd prime and x is a rational p-integer. Such congruences are concerned with the properties of p-regular functions, the congruences for h(−sp) (mod p) (s = 3, 5, 8, 12) and the sum P k≡r (mod m) p k , where h(d) is the class number of the quadratic field Q(d) of discriminant d and p-regular functions are those functions f such that f(k) (k = 0, 1, . . . ) are rational p-integers and Pn k=0 n k (−1)kf(k) ≡ 0 (mod pn) for n = 1, 2, 3, . . . We also establish many congruences for Euler numbers. MSC: Primary 11B68, Secondary 11A07, 11R29.