Riordan Pseudo-Involutions, Continued Fractions and Somos $4$ Sequences

TL;DR

This paper introduces a new family of Bell pseudo-involutions within the Riordan group, linking their generating functions to continued fractions and Somos 4 sequences, with connections to elliptic curves.

Contribution

It defines a three-parameter family of Riordan pseudo-involutions with generating functions as continued fractions, and explores their relation to Somos 4 sequences and elliptic curves.

Findings

Hankel transforms relate to Somos 4 sequences

Sequences can be associated with elliptic curves

Elliptic curves can generate Riordan pseudo-involutions

Abstract

We define a three parameter family of Bell pseudo-involutions in the Riordan group. The defining sequences have generating functions that are expressible as continued fractions. We indicate that the Hankel transforms of the defining sequences, and of the $A$ sequences of the corresponding Riordan arrays, can be associated with Somos $4$ sequence. We give examples where these sequences can be associated with elliptic curves, and we exhibit instances where elliptic curves can give rise to associated Riordan pseudo-involutions. In the case of a particular one parameter family of elliptic curves, we show how we can associate to each such curve a unique Bell pseudo-involution.