Bernoulli numbers and exact covering systems

Let b(mod a) be the arithmetic progression {n = b + caa: a E Z}. An exact covering system is a set A of (disjoint) AP's such that each integer belongs to exactly one AP. For example: B = {O(mod n), 1(mod n), ... ,(n 1)(mod n)} is an exact covering system, and when we substitute it into (2) we get (1). But, for example, {O(mod 2), 1(mod 4), 3(mod 8),... ,(2n-2 1)(mod 2n-1), (2 n-1 1)(mod 2n1)} is also an exact covering system with n AP's, and there is a superabundance of other examples [2]. If the offsets bj of an exact covering system are chosen so that 0 < bj < aj then exactly one offset is equal to zero. It will be assumed to be b1. A finite set of disjoint AP's that covers the non-negative integers, or indeed the integers from 1 to lcm{aj: 1 < j < n}, automatically covers all of the integers. For any exact cover, En=1a7 = 1. See [3]. In "A New Approach to Bernoulli Polynomials" [4], D. H. Lehmer proves that the n-th Bernoulli polynomial Bn(t) is the unique monic polynomial of degree n which satisfies Raabe's multiplication identity 1 m-i k n k: =B t + = n-mBm(nt). n k-O\ n