A generalization of rummer’s congruences

Abstract

Suppose thatm, n are positive even integers andp is a prime number such thatp-1 is not a divisor ofm. For any non-negative integerN, the classical Kummer’s congruences on Bernoulli numbersB n(n = 1,2,3,...) assert that (1-p m-1)B m/m isp-integral and (1) $$(1 - p^{m - 1} )\frac{{B_m }}{m} \equiv (1 - p^{n - 1} )\frac{{B_n }}{n}(\bmod p^{N + 1} )$$ ifm ≡ n (mod (p-1)p n). In this paper, we shall prove that for any positive integerk relatively prime top and non-negative integers α, β such that α +jk =pβ for some integerj with 0 ≤j ≤p-l.Then for any non-negative integerN, (2) $$\frac{1}{m}\{ B_m (\frac{\alpha }{k}) - p^{m - 1} B_m (\frac{\beta }{k})\} \equiv \frac{1}{n}\{ B_n (\frac{\alpha }{k}) - p^{n - 1} B_n (\frac{\beta }{k})\} (\bmod p^{N + 1} )$$ ifp-1 is not a divisor ofm andm ≡ n (mod (p-1)p n). HereB n(x) (n = 0,1,2,...) are Bernoulli polynomials. This of course contains the Kummer’s congruences. Furthermore, it contains new congruences for Bernoulli polynomials of odd indices.