[seqfan] Explicit formula for A006722
Andrew N W Hone
A.N.W.Hone at kent.ac.uk
Mon Oct 12 20:54:33 CEST 2015
Dear Seqfans,
I'm very pleased to say that after many years of thinking about Somos sequences, with my collaborator Yuri Fedorov I have finally found the explicit formula for the Somos-6 sequence, A006722<https://oeis.org/A006722>. The formula takes the form
a_{n+3} = A B^n C^{n^2 -1} \sigma (v_0 + n v) / \sigma (v)^{n^2}
where A=C / \sigma(v_0), B = A^{-1} \sigma(v) / \sigma(v_0+v) , C=i/\sqrt{20} (with i=\sqrt{-1}), \sigma is the two-variable Kleinian sigma function associated with the genus two curve
X: y^2 = 4 x^5 - 233 x^4 + 1624 x^3 - 422 x^2 + 36 x - 1,
and v and v_0 are two-component vectors in the Jacobian of X, the images under the Abel map of the divisors P_1+P_2 - 2 \infty and Q_1 + Q_2 - 2 \infty, respectively, where points P_j and Q_j on X are given by
P_1 = (-8 +\sqrt{65}, i \sqrt{13312400-1651200\sqrt{65}}),
P_2 = (-8 - \sqrt{65}, i \sqrt{13312400+1651200\sqrt{65}}),
Q_1 = (5 +2\sqrt{6}, i \sqrt{161392+65888\sqrt{6}}),
Q_2 = (5 -2\sqrt{6},-i \sqrt{161392-65888\sqrt{6}}).
With an implementation of the sigma function in Maple I have calculated approximate values A = 0.0619300608334726-0.0316569342553306*I, B = -0.0000972871331396646-0.0000157730684362499*I, v = (-.34148861398*I, .47710559600*I), v_0 = (-.37688013986-.14967709329*I, -.25910915506+.57641910658*I, but I think these are only accurate to about 4 significant figures. I'm trying to get more accurate values with a better numerical scheme.
The existence of this sort of formula was known for a long time. It is equivalent to an expression A B^n C^{n^2} \Theta (u_0 + n u) in terms of a two-variable Riemann theta function (with different constants A,B,C and two-component complex vectors u,u_0). Already in 1993 Michael Somos found a numerical fit of the sequence to such a Fourier series - see http://somos.crg4.com/somos6.html - and Noam Elkies gave a parameter-counting argument for a theta function expression in posts to Jim Propp's "robbins" forum. In fact, I proved that such a formula satisfies a Somos-6 recurrence of general type (see Analytic solutions and integrability for bilinear recurrences of order six, Andrew N.W. Hone, Applicable Analysis 89 (2010), pages 473-492), and has the correct number of parameters for any such sequence with an arbitrary (generic) choice of coefficients and initial values. So it was clear that the Somos-6 sequence A006722<https://oeis.org/A006722> must be connected to a translation on a two-dimensional complex torus, but the question still to be answered was, which translation on which torus? Now in my work with Yuri Fedorov we have shown how, for a general choice of coefficients \alpha, \beta, \gamma and initial values a_0,a_1,...,a_5 in a Somos-6 recurrence
a_{n+6}=(\alpha a_{n+5} a_{n+1} + \beta a_{n+4} a_{n+2} +\gamma a_{n+3}^2) / a_n,
one can reconstruct the shift (translation) vector v, and the initial vector v_0, on the Jacobian Jac(X) of a corresponding genus two curve X.
It turns out that there is some beautiful geometry involved in this story, which is not entirely straightforward. In order to find the curve X, one starts by obtaining a genus 4 curve S with an involution; the quotient of S by this involution is a genus 2 curve C, and the 4-dimensional torus Jac(S) splits into two 2-dimensional pieces: Jac(C) and the Prymian Prym(S). Then it happens that Prym(S) is isomorphic to Jac(X) for a certain curve X, and some recent work of A. Levin (Siegel's Theorem and the Shafarevich conjecture, J. Theor. Nombres Bordeaux 24 (2012) pages 705-727) gives an explicit recipe to find X. The other part of the story comes from the theory of integrable systems, and uses a Lax pair in terms of 3 x 3 matrices, which arises by reduction from the discrete BKP equation - aka the cube recurrence in combinatorics - and is the origin of the curve S and the shift v.
The preprint of our paper will hopefully be ready for the arXiv in the next few weeks, and in the mean time I will also prepare a more precise numerical implementation of the above formula.
Best wishes,
Andy
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