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

Hone, A.

doi:10.1090/S0002-9947-07-04215-8

Sigma function solution of the initial value problem for Somos 5 sequences
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by A. N. W. Hone PDF

Trans. Amer. Math. Soc. 359 (2007), 5019-5034 Request permission

Abstract:

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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