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Sigma function solution of the initial value problem for Somos 5 sequences. (English) Zbl 1162.11011

Summary: The Somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients. For particular choices of coefficients and initial data, integer sequences arise. By making the connection with a second order nonlinear mapping with a first integral, we prove that the two subsequences of odd/even index terms each satisfy a Somos 4 (fourth order) recurrence. This leads directly to the explicit solution of the initial value problem for the Somos 5 sequences in terms of the Weierstrass sigma function for an associated elliptic curve.

MSC:

11B37 Recurrences
33E05 Elliptic functions and integrals
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37C99 Smooth dynamical systems: general theory

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