Timeline for New binomial coefficient identity?
Current License: CC BY-SA 3.0
6 events
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| Jan 31, 2018 at 4:04 | comment | added | Zurab Silagadze | This identity also follows from the Saalschütz theorem (not immediately, but after some algebra) for the case $a=m+1/2$, $b=m+n+1$, $c=m+3/2$, because the sum now is $$\frac{\binom{n+m}{n-m}}{2m+1} {_3F_2}(m+1/2,m+n+1,-(n-m);m+3/2,2m+1;1).$$ | |
| Jan 31, 2018 at 3:17 | comment | added | Zurab Silagadze | Thanks! I obtained the identity from the Clausen’s identity for the Legendre polynomials. A generalization to the associated Legendre functions produces $$\sum\limits_{k=m}^n\frac{(-1)^{k-m}}{2k+1}\binom{n+k}{n-k}\binom{2k}{k-m}=\frac{1}{2n+1}.$$ | |
| Jan 31, 2018 at 3:07 | vote | accept | Zurab Silagadze | ||
| Jan 30, 2018 at 14:41 | history | edited | Ira Gessel | CC BY-SA 3.0 |
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| Jan 30, 2018 at 14:18 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
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| Jan 30, 2018 at 14:13 | history | answered | Ira Gessel | CC BY-SA 3.0 |