Algebraic curves, integer sequences and a discrete Painleve transcendent

TL;DR

This paper explores bilinear recurrences linked to number theory, providing explicit solutions using elliptic functions, and connects these to integrable systems and hyperelliptic curves, including applications to integer sequences and discrete Painleve equations.

Contribution

It offers explicit solutions for fourth-order bilinear recurrences via elliptic functions and relates these to integrable systems and hyperelliptic curves, extending to q-discrete Painleve equations.

Findings

Explicit solutions for recurrences using Weierstrass sigma functions.

Connection between recurrences and elliptic/divisibility sequences.

Brief exploration of q-discrete Painleve I related sequences.

Abstract

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.