Congruences involving Bernoulli polynomials

Abstract

Let { Bn (x) } be the Bernoulli polynomials. In the paper we establish some congruences for Bj (x) (mod pn), 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 ∑k ≡ r (mod m) (frac(p, k)), where h (d) is the class number of the quadratic field Q (sqrt(d)) of discriminant d and p-regular functions are those functions f such that f (k) (k = 0, 1, ...) are rational p-integers and ∑k = 0n (frac(n, k)) (- 1)k f (k) ≡ 0 (mod pn) for n = 1, 2, 3, ... . We also establish many congruences for Euler numbers. © 2007 Elsevier B.V. All rights reserved.