Algebraic curves, integer sequences and a discrete Painleve transcendent

Details Algebraic curves, integer sequences and a discrete Painleve transcendent Algebraic curves, integer sequences and a discrete Painleve transcendent

2008-07-16

arxiv.org/abs/0807.2538v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Poster at SIDE 6, Helsinki, Finland, 19-24 June 2004. One reference and some numerical values have been updated. The conjectur

Scientific paper

We consider some bilinear recurrences that have applications in number theory. The explicit solution of a general three-term bilinear recurrence relation of fourth order is given in terms of the Weierstrass sigma function for an associated elliptic curve. The recurrences can generate integer sequences, including the Somos 4 sequence and elliptic divisibility sequences. An interpretation via the theory of integrable systems suggests the relation between certain higher order recurrences and hyperelliptic curves of higher genus. Analogous sequences associated with a $q$-discrete Painlev\'e I equation are briefly considered.

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