Casting light on shadow Somos sequences

TL;DR

This paper explores shadow sequences derived from Somos-4 recurrences over dual numbers, providing explicit formulas and multiple solution perspectives, advancing the understanding of nonlinear recurrences with the Laurent property.

Contribution

It offers new explicit expressions and solution methods for shadow Somos-4 sequences over dual numbers, contributing to the development of cluster superalgebra concepts.

Findings

Explicit formulas for shadow Somos-4 sequences

Multiple solution approaches including elliptic functions and linear difference equations

Enhanced understanding of nonlinear recurrences with Laurent property

Abstract

Recently Ovsienko and Tabachnikov considered extensions of Somos and Gale-Robinson sequences, defined over the algebra of dual numbers. Ovsienko used the same idea to construct so-called shadow sequences derived from other nonlinear recurrence relations exhibiting the Laurent phenomenon, with the original motivation being the hope that these examples should lead to an appropriate notion of a cluster superalgebra, incorporating Grassmann variables. Here we present various explicit expressions for the shadow of Somos-4 sequences, and describe the solution of a general Somos-4 recurrence defined over the $\mathbb{C}$-algebra of dual numbers from several different viewpoints: analytic formulae in terms of elliptic functions, linear difference equations, and Hankel determinants.