Somos-4 equation and related equations

TL;DR

This paper explores the Somos-4 recurrence, demonstrating its connection to Gale-Robinson sequences, elliptic sequences, and solutions to the Volterra lattice, revealing new relationships among these mathematical objects.

Contribution

It proves that sequences from the Somos-4 recurrence also satisfy Gale-Robinson identities and constructs solutions to the Volterra lattice related to linear sequences.

Findings

Sequences generated by Somos-4 satisfy Gale-Robinson identities

Second-order linear sequences satisfy Somos-4 with specific coefficients

Constructed solutions to Volterra lattice related to linear sequences

Abstract

The main object of study in this paper is the well-known Somos-4 recurrence. We prove a theorem that any sequence generated by this equation also satisfies Gale-Robinson one. The corresponding identity is written in terms of its companion elliptic sequence. An example of such relationship is provided by the second-order linear sequence which, as we prove using Wajda's identity, satisfies the Somos-4 recurrence with suitable coefficients. Also, we construct a class of solutions to Volterra lattice equation closely related to the second-order linear sequence.