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

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...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2007-10, Vol.359 (10), p.5019-5034
Main Author: Hone, A. N. W.
Format: Article
Language:English
Subjects:
Algebra
Cauchy problem
Curves
Difference and functional equations, recurrence relations
Division
Elliptic functions
Exact sciences and technology
Integers
Mathematical analysis
Mathematical independent variables
Mathematical sequences
Mathematics
Multivariate analysis
Numerical analysis
Numerical analysis. Scientific computation
Partial differential equations
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Probability and statistics
Sciences and techniques of general use
Sigma function
Statistics
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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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-07-04215-8