李善兰恒等式

表达式

${\displaystyle {\binom {n+k}{k}}^{2}=\sum _{j=0}^{k}{\binom {k}{j}}^{2}{\binom {n+2k-j}{2k}}}$

与超几何函数的关系

${\displaystyle {}_{3}F_{2}(a,b,-n;c,1+a+b-c-n;1)={\frac {(c-a)_{n}(c-b)_{n}}{(c)_{n}(c-a-b)_{n}}}.}$[3] [4]

${\displaystyle \sum _{j=0}^{k}{\binom {k}{j}}^{2}{\binom {n+2k-j}{2k}}={\frac {(n+2k)!}{(2k)!n!}}\sum _{j=0}^{\infty }{\frac {(-k)^{(j)}(-k)^{(j)}(-n)^{(j)}}{(1)^{(j)}(-n-2k)^{(j)}j!}}={\frac {(n+2k)!}{(2k)!n!}}{}_{3}F_{2}(-k,-k,-n;1,-n-2k;1)}$

${\displaystyle ={\frac {(n+2k)!(1+k)_{n}(1+k)_{n}}{(2k)!n!(1)_{n}(1+2k)_{n}}}={\frac {(n+2k)!(n+k)!(n+k)!(2k)!}{(2k)!n!k!k!n!(n+2k)!}}={\binom {n+k}{k}}^{2}}$

参考资料

1. ^ 形式幂级数技巧的应用. [2013-12-10]. （原始内容存档于2019-06-09）.
2. ^ 李善兰恒等式的概率证明. [2013-12-10]. （原始内容存档于2019-06-03）.
3. ^ Yong Sup Kim and Arjun Kumar Rathie. A NEW PROOF OF SAALSCHUTZ’S THEOREM FOR THE ¨SERIES 3F2(1) AND ITS CONTIGUOUS RESULTS WITH APPLICATIONS (PDF). Commun. Korean Math. Soc. 27. 2012 [2018-06-12]. （原始内容 (PDF)存档于2018-06-12）.
4. ^ Bruce Sagan,Richard Stanley. Mathematical Essays in honor of Gian-Carlo Rota.