2024/08/31 by J. W. E. Harrow, Harrow, J. W. E., Andrew N. W. Hone +1
Computer Science · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Image and Signal Denoising Methods #Mathematical Physics (math-ph) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2409.00406
openalex publication_date 2024/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by the search for an appropriate notion of a cluster superalgebra, incorporating Grassmann variables, Ovsienko and Tabachnikov considered the extension of various recurrence relations with the Laurent phenomenon to the ring of dual numbers. Furthermore, by iterating recurrences with specific numerical values, some particular well-known integer sequences, such as the Fibonacci sequence, Markoff numbers, and Somos sequences, were shown to produce associated ``shadow'' sequences when they were extended to the dual numbers. Here we consider the most general version of the Somos-5 recurrence defined over the ring of dual numbers \mathbbD with complex coefficients, that is the ring ℂ[ε] modulo the relation ε2=0. We present three different ways to present the general solution of the initial value problem for Somos-5 and its shadow part: in analytic form, using the Weierstrass sigma function with arguments in \mathbbD; in terms of the solution of a linear difference equation; and using Hankel determinants constructed from \mathbbD-valued moments, via a connection with a Quispel-Roberts-Thompson (QRT) map over the dual numbers.