On sums of binomial coefficients and their applications

In this paper we study recurrences concerning the combinatorial sum $[n,r{]}_m={\sum }_{k\equiv r(modm)}\left(\genfrac{}{}{0px}{}{n}{k}\right)$ and the alternate sum ${\sum }_{k\equiv r(modm)}(-1{)}^{(k-r)\mathrm{/}m}\left(\genfrac{}{}{0px}{}{n}{k}\right)$, where m>0, $n\ge 0$ and r are integers. For example, we show that if $n\ge m-1$ then $$\sum _{i=0}^{\lfloor (m-1)\mathrm{/}2\rfloor }(-1{)}^i\left(\genfrac{}{}{0px}{}{m-1-i}{i}\right)[n-2i,r-i{]}_m=2^{n-m+1}\mathrm{.}$$ We also apply such results to investigate Bernoulli and Euler polynomials. Our approach depends heavily on an identity established by the author [Integers 2(2002)].