Timeline for On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"
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11 events
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| May 28, 2023 at 23:48 | comment | added | Tito Piezas III | I mean one of Zagier’s Legendrian solutions has Ramanujan-type pi formula associated with it. I’ll discuss it in my next post if that is true for all 4 Legendrian types. | |
| May 28, 2023 at 23:12 | comment | added | Tito Piezas III | I checked Zagier’s original paper. It seems he only found 3 Legendrian solutions ( #19 and two others). And the Legendrian solutions may be as notable as the sporadics as they have modular origins with Ramanujan type pi formulas. | |
| May 28, 2023 at 18:54 | comment | added | Ofir Gorodetsky | @TitoPiezasIII These are known (Zagier found 4 hypergeometric and 4 Legendrian solutions to his original recurrence, and 4+4 related solutions exist for the Almkvist-Zudilin equation), I commented under your question with references. | |
| May 28, 2023 at 17:42 | comment | added | Tito Piezas III | I found four sequences with modular origins apparently of the same cubic type found by Almkvist-Zudilin. Kindly see this post. | |
| May 22, 2023 at 19:25 | comment | added | Ofir Gorodetsky | @TitoPiezasIII Thanks for the update, I'll check out your post. | |
| May 22, 2023 at 17:56 | comment | added | Tito Piezas III | I tried to evaluate the cfrac associated with Zagier's sequence $B$. It seems to have peculiar properties such as having 6 limits, one of which has a closed-form, though I am not sure. Kindly see MO post. | |
| May 21, 2023 at 16:32 | comment | added | Tito Piezas III | Ok, I understand. Regarding Cooper's $s_7$ and $s_{10}$, the cfracs evaluate to $\zeta(2)/7$ and $\zeta(2)/5$, while Zudilin's evaluate to $\zeta(4)/13$. Would $p=13$ be a clue? It's probably nothing, but just made me wonder. | |
| May 21, 2023 at 16:26 | comment | added | Ofir Gorodetsky | I should also say that all the 15 sequences were found to have modular origins; I do not know if the 16th sequence can be explained in this way. | |
| May 21, 2023 at 16:20 | comment | added | Ofir Gorodetsky | @TitoPiezasIII I agree that there are some other sequences that may be dubbed 'sporadic', depending on your interpretation of 'sporadic' (I'll add that these specific 15 sequences were put under the same 'sporadic' umbrella by various authors; my paper did not introduce any new convention). However, each of the 15 sequences was found in a (wide but finite) search within an infinite family of recurrences, making the adjective 'sporadic' well deserved in my view. My understanding is that the 16th sequence you mention was not found by searching within a well-defined family of recurrences. | |
| May 21, 2023 at 15:17 | comment | added | Tito Piezas III | Thank you for the reference to Zagier's paper. It is good to know I got the evaluations right. You're the "O. Gorodetsky" in the post after this, I assume? By the way, I believe there are $15+1=16$ known sporadic sequences, if we include Zudilin's recurrence for $\zeta(4)$ (call it $Z$) which I discussed in that post. However, there is no Ramanujan-Sato pi formula associated with $Z$ unlike the other 15. Cooper found $t_7, t_{10}, t_{18}$, but not yet $t_{30}$ so there might still be another "sporadic sequence" out there. | |
| May 21, 2023 at 12:10 | history | answered | Ofir Gorodetsky | CC BY-SA 4.0 |