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

• Autores: A. N. W. Hone
• Localización: Transactions of the American Mathematical Society, ISSN 0002-9947, Vol. 359, Nº 10, 2007, págs. 5019-5034
• Idioma: inglés
• DOI: 10.1090/s0002-9947-07-04215-8
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• Resumen

  • 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.