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Bilinear recurrences and addition formulae for hyperelliptic sigma functions

Braden, H.W., Enolskii, V.Z., Hone, Andrew N.W. (2005) Bilinear recurrences and addition formulae for hyperelliptic sigma functions. Journal of Nonlinear Mathematical Physics, 12 (Sup.2). pp. 46-62. ISSN 1402-9251. (doi:10.2991/jnmp.2005.12.s2.5) (KAR id:41493)


The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be expressed in terms of the Weierstrass sigma function for an associated elliptic curve. Here we derive the analogous family of sequences associated with an hyperelliptic curve of genus two defined by the affine model y2=4x5+c4x4+...+c1x+c0. We show that the recurrence sequences associated with such curves satisfy bilinear recurrences of order 8. The proof requires an addition formula which involves the genus two Kleinian sigma function with its argument shifted by the Abelian image of the reduced divisor of a single point on the curve. The genus two recurrences are related to a B\"{a}cklund transformation (BT) for an integrable Hamiltonian system, namely the discrete case (ii) H\'{e}non-Heiles system.

Item Type: Article
DOI/Identification number: 10.2991/jnmp.2005.12.s2.5
Additional information: Special Issue: Symmetries and Integrability of Difference Equations SIDE VI
Subjects: Q Science > QA Mathematics (inc Computing science) > QA351 Special functions
Q Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equations
Q Science > QA Mathematics (inc Computing science) > QA564 Algebraic Geometry
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Andrew Hone
Date Deposited: 21 Jun 2014 01:28 UTC
Last Modified: 16 Nov 2021 10:16 UTC
Resource URI: (The current URI for this page, for reference purposes)

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