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 On Sat, Jun 7, 2014 at 12:29 PM, Bill Gosper <billgosper@gmail.com> wrote:
Tentative quadrivariate 𝜗₁ identity: Out[742]= EllipticTheta[1, -X + Y, q] EllipticTheta[1, X + Y, q] EllipticTheta[ 1, W - Z, q] EllipticTheta[1, W + Z, q] - EllipticTheta[1, W - Y, q] EllipticTheta[1, W + Y, q] EllipticTheta[ 1, -X + Z, q] EllipticTheta[1, X + Z, q] + EllipticTheta[1, W - X, q] EllipticTheta[1, W + X, q] EllipticTheta[ 1, -Y + Z, q] EllipticTheta[1, Y + Z, q]
(= 0?) Testing to 0ᵗʰ order: In[743]:= Series[%742 /. {X -> x*q, Y -> y*q, Z -> z*q, W -> w*q}, {q, 0, 0}]
Out[743]= O[q]^8 Nice. (Too nice? I requested essentially no terms.) OK, try 7ᵗʰ: In[745]:= Series[%742 /. {X -> x*q, Y -> y*q, Z -> z*q, W -> w*q}, {q, 0, 7}]
Out[745]= O[q]^8
Why no improvement? Trying for 8ᵗʰ, In[747]:= Series[%742 /. {X -> x*q, Y -> y*q, Z -> z*q, W -> w*q}, {q, 0, 8}] // FunctionExpand
hits a lacuna in Mma's 𝜗' knowledge: ((1/2 (-x+y)^2 (EllipticThetaPrime^(0,1,0))[1,0,0]+ <big mess>...))
So let's try random exact numerics: In[759]:= Table[v -> RandomInteger[{-9, 9}, 2].{1, I}, {v, {W, X, Y, Z}}]
Out[759]= {W -> -3 + 6 I, X -> -7 I, Y -> 6 - 6 I, Z -> -7 + 9 I}
In[760]:= Table[v -> RandomInteger[{-9, 9}, 2].{1, I}/9/Sqrt[2], {v, {q}}]
Out[760]= {q -> -((5 I)/(9 Sqrt[2]))}
In[761]:= %742 /. %% /. %
Out[761]= EllipticTheta[1, -13 + 15 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -3 - I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -3 + 13 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -1 + 3 I, -((5 I)/(9 Sqrt[2]))] - EllipticTheta[1, -9 + 12 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -7 + 2 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -7 + 16 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[1, 3, -((5 I)/(9 Sqrt[2]))] + EllipticTheta[1, -10 + 15 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[1, 4 - 3 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[1, 6 + I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[1, 6 - 13 I, -((5 I)/(9 Sqrt[2]))]
In[762]:= N[%]
Out[762]= -2.20279*10^173 + 1.54582*10^173 I
I was kinda hoping for 0. How discouraging. But wait!
In[764]:= N[%%%, 999]
During evaluation of In[764]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating EllipticTheta[1,-13+15 I,-((5 I)/(9 Sqrt[2]))] EllipticTheta[1,-3-I,-((5 I)/(9 Sqrt[2]))] EllipticTheta[1,-3+13 I,-((5 I)/(9 Sqrt[2]))] EllipticTheta[1,-1+3 I,-((5 I)/(9 Sqrt[2]))]-<<1>><<1>>EllipticTheta[1,-10+15 I,-((5 I)/(9 Sqrt[2]))] <<13>>[<<1>>] <<1>> <<1>>. >>
Out[764]= 0.*10^-856 + 0.*10^-856 I
How can people regard symbolic math as more exotic than numerics? --rwg