Factors of sums and alternating sums involving binomial coefficients and powers of integers

Details Factors of sums and alternating sums involving binomial coefficients and powers of integers Factors of sums and alternating sums involving binomial coefficients and powers of integers

2010-08-19

arxiv.org/abs/1008.3316v2

Int. J. Number Theory 7 (2011), 1959-1976

Mathematics

Number Theory

14 pages, to appear in Int. J. Number Theory

Scientific paper

10.1142/S1793042111004812

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, there holds {align*} \sum_{k=0}^{n_1}\epsilon^k (2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod (n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any nonnegative integer $r$ and positive integer $s$ such that $r+s$ is odd, $$ \sum_{k=0}^{n}\epsilon ^k (2k+1)^{r}({2n\choose n-k}-{2n\choose n-k-1})^{s} \equiv 0 \mod{{2n\choose n}}, $$ where $\epsilon=\pm 1$.

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