Casting more light in the shadows: dual Somos-5 sequences

TL;DR

This paper extends the Somos-5 recurrence relation to dual numbers, providing three methods to solve the initial value problem and exploring the associated shadow sequences with potential applications in cluster superalgebras.

Contribution

It introduces the most general form of Somos-5 over dual numbers and offers analytic, linear difference, and Hankel determinant solutions.

Findings

Explicit solutions in terms of Weierstrass sigma functions.

Connection with QRT maps over dual numbers.

Framework for shadow sequences in dual number extensions.

Abstract

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 $\mathbb{D}$ with complex coefficients, that is the ring $\mathbb{C}[\varepsilon]$ modulo the relation $\varepsilon^2=0$. 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 $\mathbb{D}$; in terms of the solution of a linear difference equation; and using Hankel determinants constructed from $\mathbb{D}$-valued moments, via a connection with a Quispel-Roberts-Thompson (QRT) map over the dual numbers.