Congruences for sums of binomial coefficients

Let q > 1 and m > 0 be relatively prime integers. We find an explicit period v(m)(q) such that for any integers n > 0 and r we have [GRAPHICS] whenever a is an integer with gcd (1 - (-a)(m), q) = 1, or a = -1 (mod q), or a 1 (mod q) and 2 vertical bar m, where [GRAPHICS] This is a further extension of a congruence of Glaisher. (C) 2007 Elsevier Inc. All rights reserved.