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22 September 2020
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On sums of binomial coefficients and their applications
Zhi-Wei Sun ;
Date 21 Apr 2004
Subject Number Theory; Combinatorics MSC-class: 11B65; 05A19; 11B37; 11B68 | math.NT math.CO
AbstractIn this paper we study recurrences concerning the combinatorial sum $S(n,r)=sum_{kequiv r (mod m)}inom {n}{k}$ and the alternate sum $sum_{kequiv r (mod m)}(-1)^{(k-r)/m}inom{n}{k}$, where $m>0$, $nge 0$ and $r$ are integers. For example, we show that if $nge m-1$ then $$sum_{i=0}^{lfloor(m-1)/2 floor}(-1)^iinom{m-1-i}i S(n-2i,r-i)=2^{n-m+1}.$$ We also apply such results to investigate Bernoulli and Euler polynomials. Our approach depends heavily on an identity given by the author [Integers 2(2002)].
Source arXiv, math.NT/0404385
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