p-adic valuations of some sums of multinomial coefficients

Zhi-Wei Sun

paper

p-adic valuations of some sums of multinomial coefficients

arXiv:0910.3892

Abstract

Let $m$ and $n > 0$ be integers. Suppose that $p$ is a prime dividing $m - 4$ but not dividing $m$ . We show that $ν_{p} (\sum_{k = 0}^{n - 1} \frac{( k 2 k )}{m ^{k}})$ and $ν_{p} (\sum_{k = 0}^{n - 1} (k n - 1) (- 1)^{k} \frac{( k 2 k )}{m ^{k}})$ are at least $ν_{p} (n)$ , where $ν_{p} (x)$ denotes the $p$ -adic valuation of $x$ . Furthermore, if $p > 3$ then $$n^{-1}\sum_{k=0}^{n-1}\frac{\bi{2k}k}{m^k}=\frac{\binom{2n-1}{n-1}}{4^{n-1}} (mod p^{ν_p(m-4)})$$ and $n^{- 1} k = 0 \sum n - 1 (k n - 1) (- 1)^{k} \frac{( k 2 k )}{m ^{k}} = \frac{C _{n - 1}}{4 ^{n - 1}} (m o d p^{ν_{p} (m - 4)}),$ where $C_{k}$ denotes the Catalan number $(k 2 k) / (k + 1)$ . This implies several conjectures of Guo and Zeng [GZ]. We also raise two conjectures, and prove that $n > 1$ is a prime if and only if $k = 0 \sum n - 1 m u l t in o mia l (n - 1) k k, ..., k = 0 (m o d n),$ where $m u l t in o mia l k_{1} + ... + k_{n - 1} k_{1}, ..., k_{n - 1}$ denotes the multinomial coefficient $(k_{1} + ... + k_{n - 1})! / (k_{1}! ... k_{n - 1}!)$ .

16 pages

p-adic valuations of some sums of multinomial coefficients

Abstract

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