Proof of the Somos-4 Hankel Determinants Conjecture

TL;DR

This paper proves Somos's conjecture that the Hankel determinants of a specific generating series are the Somos-4 numbers, using quadratic transformations for Hankel determinants.

Contribution

It provides a rigorous proof of the Somos-4 Hankel determinants conjecture employing quadratic transformations, advancing understanding of these special sequences.

Findings

Confirmed that the Hankel determinants match Somos-4 numbers

Validated the use of quadratic transformations in this context

Established a new proof technique for such conjectures

Abstract

By considering the fundamental equation $x=y-y^2=z-z^3$, Somos conjectured that the Hankel determinants for the generating series $y(z)$ are the Somos-4 numbers. We prove this conjecture by using the quadratic transformation for Hankel determinants of Sulanke and Xin.