On sums of binomial coefficients and their applications

Abstract

In this paper we study recurrences concerning the combinatorial sum $[n,r{]}_m={\sum }_{k\equiv r(modm)}(\genfrac{}{}{0ex}{}{n}{k})$ and the alternate sum ${\sum }_{k\equiv r(modm)}(-1{)}^{(k-r)/m}(\genfrac{}{}{0ex}{}{n}{k})$, where m>0, $n\ge 0$ and r are integers. For example, we show that if $n\ge m-1$ then $$\sum _{i=0}^{\lfloor (m-1)/2\rfloor }(-1{)}^i(\genfrac{}{}{0ex}{}{m-1-i}{i})[n-2i,r-i{]}_m=2^{n-m+1}.$$ 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)].

References in corpus (1)

• An extension of a curious binomial identity