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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: A. N. W. Hone
Journal: Trans. Amer. Math. Soc. 359 (2007), 5019-5034
MSC (2000): Primary 11B37, 33E05; Secondary 37J35
Published electronically: April 24, 2007
MathSciNet review: 2320658
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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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Additional Information

A. N. W. Hone
Affiliation: Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, United Kingdom
Email: anwh@kent.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04215-8
PII: S 0002-9947(07)04215-8
Keywords: Integer sequences, elliptic curves
Received by editor(s): February 9, 2005
Received by editor(s) in revised form: September 15, 2005
Published electronically: April 24, 2007
Article copyright: © Copyright 2007 American Mathematical Society