Congruences for sums of binomial coefficients

Abstract

Let q>1 and m>0 be relatively prime integers. We find an explicit period ${\nu }_m(q)$ such that for any integers n>0 and r we have $[n+{\nu }_m(q),r{]}_m(a)=[n,r{]}_m(a)(modq)$ whenever a is an integer with $\gcd (1-(-a{)}^m,q)=1$, or a=-1 (mod q), or a=1 (mod q) and 2|m, where $[n,r{]}_m(a)={\sum }_{k=r(modm)}(\genfrac{}{}{0ex}{}{n}{k})a^k$. This is a further extension of a congruence of Glaisher.