On sums of binomial coefficients and their applications

In this paper we study recurrences concerning the combinatorial sum nrm=∑k≡r(modm)nk and the alternate sum ∑k≡r(modm)(-1)(k-r)/mnk, where m>0m>0, n⩾0n⩾0 and r   are integers. For example, we show that if n⩾m-1n⩾m-1 then∑i=0⌊(m-1)/2⌋(-1)im-1-iin-2ir-im=2n-m+1.We also apply such results to investigate Bernoulli and Euler polynomials. Our approach depends heavily on an identity established by the author [A curious identity involving binomial coefficients, Integers: Electron. J. Combin. Number Theory 2 (2002), A4, 8pp].