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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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.
References
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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
  • Received by editor(s): February 9, 2005
  • Received by editor(s) in revised form: September 15, 2005
  • Published electronically: April 24, 2007
  • © Copyright 2007 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 5019-5034
  • MSC (2000): Primary 11B37, 33E05; Secondary 37J35
  • DOI: https://doi.org/10.1090/S0002-9947-07-04215-8
  • MathSciNet review: 2320658