Ideally, we might find formulas for a(x), c(x), d(x), and q(x). (I keep providing for b(x), but it always comes out identically 1. ?) On Mon, Jun 9, 2014 at 4:30 AM, Bill Gosper <billgosper@gmail.com> wrote:
Rejected but juicy http://arxiv.org/abs/math/0703470 (p 9) describes the remarkable (to me, anyway) *polynomial* valued Somos4
s[0] = 0; s[1] = s[2] = 1; s[3] = -1; s[4] = x; s[n_Integer /; n > 4] := Factor[(s[n - 1]*s[n - 3] + s[n - 2]^2)/s[n - 4]]
In[536]:= s /@ Range[12]
Out[536]= {1, 1, -1, x, 1 + x, -1 - x + x^2, -1 - x - x^3, -x (2 + 3 x), 1 + 3 x + 3 x^2 - x^4 + x^5, -(1 + x) (-1 - 2 x + 2 x^2 + 3 x^3 - x^4 + x^5), -1 - 3 x - 3 x^2 - 5 x^3 - 9 x^4 - 3 x^5 + 2 x^6 - x^7, -x (-1 - x + x^2) (3 + 9 x + 9 x^2 + 5 x^3 + 2 x^4 + x^6)}
and then on p10 "Besides the EDS condition, we retain the #1 three-variable identity s_2 j s_k s_n s_n+k = s_j s_k−j s_n−j s_n+k+j + s_j s_k+j s_n+j s_n+k−j This can be subscript-balanced as s_2 j s_k+j s_−n−k−j s_n = s_−j s_−k s_j−n s_n+k+2 j − s_−j s_k+2 j s_−n−k s_n+j , but its asymmetry and failure to subsume the E[lliptic]D[ivisibility]S[equence] condition suggest that we’re missing a nice, four-variable relation."
In fact, the 𝜗₁ identity below, which becomes s[j - m] s[j + m] s[-k + n] s[k + n] == s[-k + m] s[k + m] s[j - n] s[j + n] + s[j - k] s[j + k] s[-m + n] s[m + n] .
With something like the first seven values, this serves as an alternate definition of s[n].
For x:=1, s[n] can be expressed in "closed form": a*c^n^2*EllipticTheta[1, d*n, q]: (0.31749282989638009698851538146011901061695 + 0.41577568158982458340525424882529254242763 I) (0.74320667986312908167383113199924669418636 - 0.69294655945321371719182977376188612597500 I)^n^2 EllipticTheta[ 1, (1.7554385915026838183474896476936322715588657 + 0.050402131298346764198930943819803546234487011 I) n, -0.43035475675354998492420504350604355525329714 - 0.63418111840450730747740547053917541440014778 I]
Table[%, {n, -4, 13}] // Chop
{-1.000000000000000000000000000000000000000, 1.0000000000000000000000000000000000000000, -1.0000000000000000000000000000000000000000, -1.0000000000000000000000000000000000000000, 0, 1.0000000000000000000000000000000000000000, 1.0000000000000000000000000000000000000000, -1.0000000000000000000000000000000000000000, 1.000000000000000000000000000000000000000, 2.00000000000000000000000000000000000000, -1.000000000000000000000000000000000000000, -3.00000000000000000000000000000000000000, -5.00000000000000000000000000000000000000, 7.0000000000000000000000000000000000000, -4.0000000000000000000000000000000000000, -23.000000000000000000000000000000000000, ...}
(Find *those* a,c,d,q in ISC.) The paper suggests that there are several other such expressions for s[n]. It will be interesting to see how a, c, d, and q vary with x. --rwg
It's a bit tedious, crawling around in (4D) d,q space with FindRoot, which *loves* to numerically crash 𝜗 by running |q|>1. Or fail to converge for too large a step size. Or most amusing of all, sidestep a nasty spot and then crawl back to the original x, getting different a,c,d,q, with Riemann laughing behind my back. An ambitious exploration of this space would probably produce some interesting graphics, but not much in the way of formulas. (Note that the paper gives closed forms for a few algebraic x.) --rwg This got revived when Rich privately remarked that sets of rational points on elliptic curves might follow a somosoid rule. [Clipped: stuff on four variable theta relation]