Timeline for Linear eta product identities - how many are there?
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7 events
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| Feb 2, 2014 at 20:34 | comment | added | Wolfgang | Of all the ones just mentioned, only a few can be written in terms of $a_i$'s and $b_i$'s for appropriate $P,Q$: $I_{240}$ with $P,Q=3,5$ becomes (there is only one way to do it) $a_8a_1b_8-a_1a_4b_{16}+a_4a_{16}b_1-a_{16}a_2b_2=0$. Note that the indices are sort of "multiplicatively cyclic modulo 32" here. For the ones with products of four of level 300 and 450, it is also possible but lengthy. (Nice patterns, though.) For $I_{252}$ it is almost possible with $P,Q=7,9$. | |
| Feb 2, 2014 at 18:50 | comment | added | Wolfgang | There are some more linear ones, 14 altogether in Somos' collection, some of them with products of four, but many terms (see the 2 last ones in Somos’ level 300 file, the two of level 450 and the level 945 one), others with products of six (to wit, two pairs for each 180 & 300, one for 252 somewhat similar to $I_{60}$, and one for 240). Somos has searched in vain for linear ones with products of eight. They all have many internal symmetries, as may be expected. E.g. the pairs for 180 and 300 are perfectly "isomorphic" to each other: switch all factors 3 with factors 5. Interesting! | |
| Feb 1, 2014 at 14:32 | comment | added | Wolfgang | @TitoPiezasIII Your notation $a_k$ and $b_k$ is better than mine (which only handles $k$'s that are powers of 2), moreover it shows that for $I_{60}$ and $I_{210}$, all terms $a_kb_\ell$ have $k\ell=const$. I agree with you, that seems to cry for generalization, but in between I have gained the impression that in spite of the thousands of existing eta-identities, everything is finite there in terms of re-occuring patterns. | |
| Feb 1, 2014 at 1:15 | comment | added | Tito Piezas III | I tried $P,Q = 11,13$. Since $LCM(11\cdot12\cdot13) =1726$ (which has 24 divisors), I hoped to find linear relations between the 6 real numbers $a_1 b_{12},\, a_2 b_6,\, a_3 b_4,\, a_4 b_3,\, a_6 b_2,\, a_{12} b_1$. Unfortunately, Mathematica couldn't seem to find anything. Sigh. | |
| Feb 1, 2014 at 1:13 | comment | added | Tito Piezas III | If we define $a_k =\eta(q^k)\,\eta(q^{PQk})$ and $b_k= \eta(q^{Pk})\,\eta(q^{Qk})$, then for $P,Q =3,5$ we have $I_{60} \iff a_1b_4+a_4b_1 = a_2b_2$. For $P,Q =5,7$, it is $I_{210} \iff a_1b_6+a_3b_2 = a_2b_3+a_6b_1$. It's so tempting to speculate that these belong to an infinite family for appropriately chosen primes $P,Q$. | |
| Mar 20, 2012 at 7:23 | comment | added | Alexander Chervov | Kyoji Saito has papers on various eta product identities... I am not sure this is relevant, just a comment. | |
| Mar 19, 2012 at 22:47 | history | asked | Wolfgang | CC BY-SA 3.0 |