Sigma function solution of the initial value problem for Somos 5 sequences

Details Sigma function solution of the initial value problem for Somos 5 sequences Sigma function solution of the initial value problem for Somos 5 sequences

2005-01-31

arxiv.org/abs/math/0501554v3

Mathematics

Number Theory

More typos corrected, and extra Proposition 2.5 added. To appear in Trans. Amer. Math. Soc

Scientific paper

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.

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