Timeline for On four Ramanujan-type "Legendrian" sequences with a 3-term recurrence?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Sep 19 at 2:24 | vote | accept | Tito Piezas III | ||
| Aug 27 at 22:38 | answer | added | Ofir Gorodetsky | timeline score: 5 | |
| May 29, 2023 at 13:55 | history | edited | Tito Piezas III | CC BY-SA 4.0 |
Changed title.
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| May 29, 2023 at 13:36 | comment | added | Tito Piezas III | @OfirGorodetsky: I just read the Almkvist-Zudilin paper and in Section 4, they mentioned that Zagier found 4 Legendrian solutions in 2009. I looked at Zagier's 2009 paper and in his table of 36 solutions, he explicitly says, and I quote, "three triples", namely \begin{align} &\text{No.}19,\; (2\times 16,\; 16^2,\; 12)\\ &\text{No.}25,\; (2\times 27,\; 27^2,\; 21)\\ &\text{No.}26,\; (2\times 16,\; 16^2,\; 28)\end{align} I believe he didn't find $64^2$ and $432^2$ as those were beyond the search range. When I saw those numbers, I immediately checked $64^2$ and $432^2$, and they worked. | |
| May 29, 2023 at 12:10 | comment | added | Ofir Gorodetsky | @TitoPiezasIII Sure, will do this later this week due to busyness... | |
| May 29, 2023 at 11:55 | history | edited | Tito Piezas III | CC BY-SA 4.0 |
Added eta(5)
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| May 29, 2023 at 5:47 | history | edited | Tito Piezas III | CC BY-SA 4.0 |
Changed some details.
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| May 29, 2023 at 0:17 | history | edited | Tito Piezas III | CC BY-SA 4.0 |
Phrasing changes
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| May 28, 2023 at 23:17 | comment | added | Tito Piezas III | @OfirGorodetsky Can you convert your comments as an answer so i can upvote it? You’ve essentially solved 2/3 of the questions. 😊 | |
| May 28, 2023 at 21:36 | answer | added | Henri Cohen | timeline score: 7 | |
| May 28, 2023 at 20:03 | comment | added | Henri Cohen | As requested: $F_1=0.00457936356455642911364499027248614471814419210576351311808$, $F_2=0.0414268220026378096205680227378000709249113690570134530631$, $F_3=0.136756232683283246080196823032970017605464751401410734953$. On the other hand $F_4$ converges like $A-1/\log(n)$, so I am unable to compute $A$, which is approximately $0.51$. | |
| May 28, 2023 at 18:53 | comment | added | Ofir Gorodetsky | I do not know about question 3. | |
| May 28, 2023 at 18:51 | comment | added | Ofir Gorodetsky | In Theorem 4.1 of that paper, a correspondence is established between solutions to Zagier's equation and the Almkvist-Zudilin equation, which explains why your 4 solutions exist. They are also described in equations (4.9)-(4.11). A binomial formula (that follows from these equations) is given in Q4 in Section 5 here: arxiv.org/abs/2102.11839 . This partially answers 2: there is a unified formula for all 4 sequences as $C_{\alpha}^n \sum_{k=0}^{n} \binom{-\alpha}{n-k}^2 \binom{\alpha-1}{k}^2$, $C_{\alpha} = \alpha^{-3}$ or $2\alpha^{-3}$ and $\alpha \in \{1/2, 1/3, 1/4, 1/6\}$. | |
| May 28, 2023 at 18:46 | comment | added | Ofir Gorodetsky | Zagier found 4 hypergeometric and 4 Legendrian solutions for his recurrences. In section 4 of "Generalizations of Clausen's formula and algebraic transformations of Calabi-Yau differential equations" by Almkvist, van Straten and Zudilin, the authors describe the hypergeometric and Legendrian solutions to Zagier's equation (see equations (4.1)-(4.7)) and the connection between the hypergeometric and the Legendrian ones. Your constants 16, 27, 64 and 432 appear explicitly in (4.2). | |
| May 28, 2023 at 18:43 | comment | added | Ofir Gorodetsky | The answer to 1 is 'yes'. Before I explain why, note that the "HolonomicFunctions" Mathematica package package can quickly verify that the integer-valued $\alpha(n),\beta(n),\gamma(n),\delta(n)$ indeed satisfy your recurrences, answering 1 in the positive. | |
| May 28, 2023 at 17:48 | history | edited | Tito Piezas III | CC BY-SA 4.0 |
Corrected small typo
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| May 28, 2023 at 17:33 | history | asked | Tito Piezas III | CC BY-SA 4.0 |