A sufficient condition for $(\alpha ,\beta )$ Somos $4$ Hankel determinants

Abstract

By using Sulanke-Xin continued fractions method, Xin proposed a recursion system to solve the Somos 4 Hankel determinant conjecture. We find Xin's recursion system indeed give a sufficient condition for $(\alpha ,\beta )$ Somos $4$ sequences. This allows us to prove 4 conjectures of Barry on $(\alpha ,\beta )$ Somos $4$ sequences in a unified way.