On Mon, Jun 9, 2014 at 7:36 PM, Bill Gosper <billgosper@gmail.com> wrote:
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)}
[Clipped: four variable relations] The polynomial degree goes up like n^2/16 plus a period 8 ripple.
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
[Clipped: Numeric difficulties.]
For x = -2/3, we get a slightly messy but exact closed form 3^(1/16 (-n^2 + (5 + (-1)^n - 2 Sqrt[2] Cos[(n π)/4] + 4 Cos[(n π)/2])^2 Sin[(n π)/4]^4)) (4 Cos[(n π)/ 8] Cos[(n π)/4] - (-1)^n (Sqrt[2] + (-2 + Sqrt[2]) Cos[(n π)/4]) Cos[(n π)/ 2]) Sin[(n π)/8] (no 𝜗, no mysterious q), but 0,1,1,-1,-2/3,1/3,1/3^2,-1/3^3,0,1/3^5,-1/3^6,-1/3^7,2/3^9,1/3^10,... is no longer a Somos sequence! The problem is that s[8n] is a multiple of 3x+2, and there's no way to continue the recurrence more than three steps past those periodic 0s. The paper's "closed form" had an eight way case statement. There is also an exact 𝜗 expression ((-1)^(1/8) Sqrt[2] EllipticTheta[2, 0, I q])/(3^(1/4) EllipticTheta[2, 0, q]) where q ->Root[{-((1 + I)/3^(1/4)) + ( QPochhammer[-1, #1^2] QPochhammer[-(1/#1^2), #1^4])/( 2 (1 + 1/#1^2)) &, 0.591308037470 + 0.442317013236 I}] specifying q, and thus the 𝜗 quotient, to arbitrary precision, but with no hint the values are rational, or even real. Interestingly, the problematic (0/0) values of the recurrence are s[8n+4], whereat the 𝜗 quotient is conveniently independent of q. The paper mentions finding q -> Root[{Product[(1 + #^(2*k))*(1 + #^(4*k - 2)), {k, \[Infinity]}] - (1 + I)/ 3^(1/4) &, .7241830710727415040344246937315* I + .5068861260317593704061905537186}] producing a puzzling algebraic sequence, but Root can't pick up the scent. More generally, if there is some algebraic x for which s[n] vanishes, then it will periodically vanish for s[k n], and there will be a messy (non-𝜗) closed form, as above. --rwg