arXiv math/05096482005-09math.COmath.NT

一个组合恒等式及其对Catalan数的应用

A combinatorial identity with application to Catalan numbers

Hao Pan, Zhi-Wei Sun

备注：: 22 pages
发表信息：: Discrete Math. 306(2006), 1921-1940

摘要：

通过一个非常简单的论证，我们证明了：如果$l,m,n$是非负整数，则$ $\sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m} =\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}. On the basis of this identity, for $ d，r= 0，1，2，. $ we construct explicit $ F（d，r）$ and $ G（d，r）$ such that for any prime $ p> \max \{d，r\} $ we have \sum_{k=1}^{p-1}k^r C_{k+d}\equiv \cases F(d,r)(mod p)& if 3|p-1, \\G(d,r)\ (mod p)& if 3|p-2, where $ C_n $ denotes the Catalan number $（n+1）^{-1} \binom{2n}{n} $. For example, when $ p \geq 5$是素数，我们有\sum_{k=1}^{p-1} k^2C_k \equiv \cases-2/3（mod p）&如果3| p-1，\1/3（mod p）& if 3| p-2;和\sum_{0<k<p-4} \frac{C_{k+4}} k \equiv \cases 503/30（mod p）&如果3| p-1，-100/3（mod p）& if 3| p-2。本文还包含了Catalan数的一些新的递推关系。

英文摘要：

By a very simple argument, we prove that if $l,m,n$ are nonnegative integers then $$\sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m} =\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}. On the basis of this identity, for $d,r=0,1,2,...$ we construct explicit $F(d,r)$ and $G(d,r)$ such that for any prime $p>\max\{d,r\}$ we have \sum_{k=1}^{p-1}k^r C_{k+d}\equiv \cases F(d,r)(mod p)& if 3|p-1, \\G(d,r)\ (mod p)& if 3|p-2, where $C_n$ denotes the Catalan number $(n+1)^{-1}\binom{2n}{n}$. For example, when $p\geq 5$ is a prime, we have \sum_{k=1}^{p-1}k^2C_k\equiv\cases-2/3 (mod p)& if 3|p-1, \1/3 (mod p)& if 3|p-2; and \sum_{0<k<p-4}\frac{C_{k+4}}k \equiv\cases 503/30 (mod p)& if 3|p-1, -100/3 (mod p)& if 3|p-2. This paper also contains some new recurrence relations for Catalan numbers.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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