[math-fun] Close enough for Government work
Tentative quadrivariate πβ 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 0α΅Κ° 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 7α΅Κ°: 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 8α΅Κ°, 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 π' 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
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_kβj s_nβj s_n+k+j + s_j s_k+j s_n+j s_n+kβj This can be subscript-balanced as s_2 j s_k+j s_βnβkβj s_n = s_βj s_βk s_jβn s_n+k+2 j β s_βj s_k+2 j s_βnβk s_n+j , but its asymmetry and failure to subsume the E[lliptic]D[ivisibility]S[equence] condition suggest that weβre missing a nice, four-variable relation." In fact, the πβ 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 πβ 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 0α΅Κ° 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 7α΅Κ°: 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 8α΅Κ°, 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 π' 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
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_kβj s_nβj s_n+k+j + s_j s_k+j s_n+j s_n+kβj This can be subscript-balanced as s_2 j s_k+j s_βnβkβj s_n = s_βj s_βk s_jβn s_n+k+2 j β s_βj s_k+2 j s_βnβk s_n+j , but its asymmetry and failure to subsume the E[lliptic]D[ivisibility]S[equence] condition suggest that weβre missing a nice, four-variable relation."
In fact, the πβ 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 π 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]
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 Ο)/4] + 4 Cos[(n Ο)/2])^2 Sin[(n Ο)/4]^4)) (4 Cos[(n Ο)/ 8] Cos[(n Ο)/4] - (-1)^n (Sqrt[2] + (-2 + Sqrt[2]) Cos[(n Ο)/4]) Cos[(n Ο)/ 2]) Sin[(n Ο)/8] (no π, 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 π 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 π 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 π 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-π) closed form, as above. --rwg
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Bill Gosper