Timeline for Closed form for $\sum\limits_{M=2}^{\infty}\sum\limits_{n=M}^{\infty} \frac{2^{-M}3^{M-n-1}\pi \csc(n \pi) \Gamma(M)}{(n+1)^2 \Gamma(M-n)\Gamma(n+1)}$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jun 1 at 18:16 | comment | added | David H | @Srini $S_{n,p}$ is commonly known as the Nielsen generalized polylogarithm. It's a useful intermediate function for reducing expressions to standard polylogarithms. I found these notes extremely helpful when I was first learning about polylog integrals: people.sc.fsu.edu/~dduke/devoto-duke-84.pdf | |
| Jun 1 at 18:15 | comment | added | Srini | Unfortunately, I just realized it is a known result following basic dilogarithm formulas. So there it goes :( | |
| Jun 1 at 17:57 | comment | added | Srini | ... I get a very nice relationship between $Li_{2}(\frac23)$ and $Li_{2}(-\frac12)$. I can't tell if it is common knowledge for mathematicians. From me looking up known dilogarithms here(functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/03/…), I can't find such a relationship. In case what I stated is unexpected, please let me know. I will post the details in a dedicated post. By brute force, I don't mean numeric methods. These are fully analytical solutions. | |
| Jun 1 at 17:53 | comment | added | Srini | The part I am not clear about is the function $S_{1,2}$. I don't know how to search for the definition of that in wolfram. You'll laugh if you know about the context. I am a math-enthusiast with no formal math education. But that doesn't prevent me from dreaming up solving big math problems (LOL). These double summations are coming from me attempting to find closed forms for some PolyLog(constants) by brute force. For example, the 2 giant summations you solved (in this and my previous question) are part of a very elaborate way to attempt closed forms. So far, after you solved these, contd... | |
| Jun 1 at 17:31 | history | bounty ended | Srini | ||
| Jun 1 at 17:31 | vote | accept | Srini | ||
| Jun 1 at 14:28 | comment | added | David H | @Srini You're very kind. I feel like my answers tend to be too long-winded to be called beautiful. If there is any step in particular that you think needs clarifying, let me know. I'm also curious about the context of these double series. Did they arise in some application, or are they problems in a book somewhere? | |
| Jun 1 at 4:19 | comment | added | Srini | I am speechless. I can only admire the beauty of your technique knowing I was nowhere close to a result with my attempts. Will try to understand your steps carefully and accept your answer tomorrow. Happy your numeric value matches. | |
| Jun 1 at 3:44 | history | answered | David H | CC BY-SA 4.0 |