查看“︁李善兰恒等式”︁的源代码

'''李善兰恒等式'''为[[组合数学]]中的一个[[恒等式]]，由[[中国]][[清代]][[数学家]][[李善兰]]于1859年在《垛积比类》一书中首次提出，因此得名。

有[[幂级数]]<ref>{{cite web | title =形式幂级数技巧的应用 | url =http://www.cnki.com.cn/Article/CJFDTotal-SSJS199001014.htm | access-date =2013-12-10 | archive-date =2019-06-09 | archive-url =https://web.archive.org/web/20190609004838/http://www.cnki.com.cn/Article/CJFDTotal-SSJS199001014.htm | dead-url =no }}</ref>和[[概率]]<ref>{{cite web | title =李善兰恒等式的概率证明 | url =http://www.cnki.com.cn/Article/CJFDTotal-XUSJ200604042.htm | access-date =2013-12-10 | archive-date =2019-06-03 | archive-url =https://web.archive.org/web/20190603205605/http://www.cnki.com.cn/Article/CJFDTotal-XUSJ200604042.htm | dead-url =no }}</ref>两种证明方法。

==表达式==
<math>{\binom {n+k}k}^2=\sum_{j=0}^k {\binom kj}^2 \binom {n+2k-j}{2k}</math>

其中<math>{\binom {k}l}=\frac{k!}{l!(k-l)!}</math>

==与超几何函数的关系==
'''李善兰恒等式'''是[[广义超几何函数|薩爾許茨定理]]（Saalschütz's theorem）的一个整数特例。

<math>{}_3F_2 (a,b, -n;c, 1+a+b-c-n;1)= \frac{(c-a)_n(c-b)_n}{(c)_n(c-a-b)_n}.</math><ref>{{cite journal|journal=Commun. Korean Math. Soc. 27|year=2012|author=Yong Sup Kim and Arjun Kumar Rathie|title=A NEW PROOF OF SAALSCHUTZ’S THEOREM FOR THE ¨SERIES 3F2(1) AND ITS CONTIGUOUS RESULTS WITH APPLICATIONS|url=http://www.mathnet.or.kr/mathnet/thesis_file/CKMS-27-1-129-135.pdf|access-date=2018-06-12|archive-url=https://web.archive.org/web/20180612162609/http://www.mathnet.or.kr/mathnet/thesis_file/CKMS-27-1-129-135.pdf|archive-date=2018-06-12|dead-url=yes}}</ref>
<ref>{{cite book|author=Bruce Sagan,Richard Stanley|title=Mathematical Essays in honor of Gian-Carlo Rota}}</ref>

<math>\sum_{j=0}^k \binom{k}{j}^2\binom{n+2k-j}{2k}
=\frac{(n+2k)!}{(2k)!n!}\sum_{j=0}^{\infty} \frac{(-k)^{(j)}(-k)^{(j)}(-n)^{(j)}}{(1)^{(j)}(-n-2k)^{(j)}j!}
=\frac{(n+2k)!}{(2k)!n!}{}_3F_2 (-k,-k,-n;1,-n-2k;1)</math>

<math>=\frac{(n+2k)!(1+k)_n(1+k)_n}{(2k)!n!(1)_n(1+2k)_n}
=\frac{(n+2k)!(n+k)!(n+k)!(2k)!}{(2k)!n!k!k!n!(n+2k)!}=\binom{n+k}{k}^2</math>

==参见==
*[[组合数学]]
*[[李善兰]]

==参考资料==
{{reflist}}

{{中国数学史}}

[[Category:组合数学]]
[[Category:中国科技]]
{{algebra-stub}}