Congruences for sums of binomial coefficients

AbstractLet q>1 and m>0 be relatively prime integers. We find an explicit period νm(q) such that for any integers n>0 and r we have[n+νm(q)r]m(a)≡[nr]m(a)(modq) whenever a is an integer with gcd(1−(−a)m,q)=1, or a≡−1(modq), or a≡1(modq) and 2|m, where [nr]m(a)=∑k≡r(modm)(nk)ak. This is a further extension of a congruence of Glaisher