Timeline for Proving a binomial sum identity
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Jul 27, 2021 at 18:57 | vote | accept | T. Amdeberhan | ||
| Jul 27, 2021 at 17:24 | history | edited | T. Amdeberhan | CC BY-SA 4.0 |
added 167 characters in body
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| Jul 27, 2021 at 13:26 | comment | added | T. Amdeberhan | That's also an interesting proposition. | |
| Jul 27, 2021 at 5:43 | comment | added | Alexander Burstein | @T.Amdeberhan I wonder if there is a nice probabilistic interpretation of $$\sum_{n=0}^{\infty}{\frac{\binom{2n}{n}\binom{2x}{x}x}{2^{2(n+x)}(n+x)}}=1.$$ | |
| Jul 26, 2021 at 21:04 | comment | added | T. Amdeberhan | @AlexanderBurstein: thanks, this is a neater form. | |
| Jul 26, 2021 at 13:28 | comment | added | T. Amdeberhan | @Nemo: One of my tags is "soft question". Nonetheless, look at the rich and pedagogical responses. | |
| Jul 26, 2021 at 13:23 | comment | added | Ira Gessel | The sum is $x^{-1}(\,{}_2F_1(1/2,𝑥;1+𝑥\mid 1)−1)$, which can be evaluated by Gauss's theorem, one of the standard hypergeometric summation theorems. Max Alekseyev's proof is essentially a special case of the usual proof of Gauss's theorem. | |
| Jul 26, 2021 at 10:41 | review | Close votes | |||
| Jul 29, 2021 at 7:35 | |||||
| Jul 26, 2021 at 10:28 | comment | added | Nemo | This is elementary textbook material. See exercise 8 to chapter 12 of Whittaker and Watson. | |
| Jul 26, 2021 at 9:37 | answer | added | Alapan Das | timeline score: 4 | |
| Jul 26, 2021 at 8:08 | history | edited | Martin Sleziak |
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| Jul 26, 2021 at 3:44 | comment | added | Alexander Burstein | This can be rewritten as $$\sum_{n=0}^{\infty}\frac{\binom{2n+2}{n+1}}{2^{2n+2}\,(n+x+1)}=\frac{2^{2x}}{x\,\binom{2x}x}-\frac{1}{x},$$ or equivalently, $$\sum_{n=0}^{\infty}\frac{\binom{2n}{n}}{2^{2n}\,(n+x)}=\frac{2^{2x}}{x\,\binom{2x}x}.$$ | |
| Jul 26, 2021 at 0:42 | history | became hot network question | |||
| Jul 25, 2021 at 22:52 | answer | added | Max Alekseyev | timeline score: 9 | |
| Jul 25, 2021 at 17:41 | answer | added | user127776 | timeline score: 6 | |
| Jul 25, 2021 at 17:24 | answer | added | Carlo Beenakker | timeline score: 3 | |
| Jul 25, 2021 at 16:37 | history | asked | T. Amdeberhan | CC BY-SA 4.0 |