Timeline for Are these two $q$-continued fractions equivalent?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Sep 19, 2023 at 16:47 | comment | added | Wolfgang | You are right. :) Those simple identities can be fascinating, e.g., $$\sqrt{\frac{ \sqrt {65} +7}{8}}+\sqrt{\frac{ \sqrt {65}-1 }{8}} = \sqrt{ \frac{\sqrt{5} + 1}{2} \cdot \frac{\sqrt{13} + 3}{2} }$$ | |
| Sep 19, 2023 at 8:54 | vote | accept | Tito Piezas III | ||
| Sep 19, 2023 at 1:35 | comment | added | Tito Piezas III | @Wolfgang Thanks. However, I also retained the old form as two forms can give exotic identities. For example, equating the two, we get, $$2\sqrt{3\big(3+\sqrt3-3^{3/4}\sqrt{2+\sqrt3}\big)}+ \sqrt{2}( 3+ \sqrt{2}) - \sqrt[4]{3}\,(3+\sqrt{3})=2$$ | |
| Sep 19, 2023 at 1:14 | history | edited | Tito Piezas III | CC BY-SA 4.0 |
Alternative forms
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| Sep 18, 2023 at 9:53 | comment | added | Wolfgang | I just simplified the last fraction, it looks a bit less intimidating now. :) | |
| Sep 18, 2023 at 9:52 | history | edited | Wolfgang | CC BY-SA 4.0 |
simplified the last fraction, much less ugly now
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| Aug 16, 2023 at 13:41 | answer | added | Nicco | timeline score: 6 | |
| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
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| Dec 14, 2015 at 4:31 | comment | added | Tito Piezas III | @NoamD.Elkies: Can you kindly look at this answer and see if my Part 2 is correct? (Re $\tau$ as quadratic vs quartic roots.) | |
| Oct 3, 2015 at 23:09 | comment | added | ccorn | Related answer together with the posts linked therein. You may want to flip the signs of $a$ and $q$ (here) for easier comparison. The condition $ab=q$ is preserved under that substitution. To summarize: Yes, and Nicco's cfrac is an instance of entry 11 (not 12) that for $ab=q$ can be simplified to a theta quotient like entry 12. I suppose Ramanujan was aware of that, but sought a cfrac that was not restricted to $ab=q$, and entry 12 was the result. I hesitate to make an answer out of this, as I have already written four related posts on MSE. | |
| Aug 19, 2015 at 5:09 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Added infinite product.
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| Jul 7, 2015 at 4:43 | comment | added | Tito Piezas III | @NoamD.Elkies: Yes, I was not being precise. :) | |
| Jul 7, 2015 at 3:54 | comment | added | Noam D. Elkies | Presumably algebraic numbers only at special values of $q$ (namely $e^{2\pi i z}$ where $z$ is a quadratic irrationality in the upper half-plane; e.g. $z=i$ gives your $e^{-2\pi}$). | |
| Jul 7, 2015 at 1:22 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Clarified general form.
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| Jul 6, 2015 at 15:43 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
added 17 characters in body
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| Jul 6, 2015 at 15:36 | history | asked | Tito Piezas III | CC BY-SA 3.0 |