On sums of binomial coefficients and their applications

In this paper we study recurrences concerning the combinatorial sum n r m = kr (mod m) n k and the alternate sum kr (mod m)(-1)(k-r)/m n k , where m > 0, n 0 and r are integers. For example, we show that if n m - 1 then (m-1)/2 i=0 (-1)i m - 1 - i i n - 2i r - i m = 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 [Integers 2(2002)].

• Type: Journal
• Year: 2008
• Where: DM
• Authors: Zhi-Wei Sun
• Keywords: Alternate Sum Kr, Combinatorial Sum, DM 2008, Recurrences