Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials

TL;DR

This paper explores the connections between generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, revealing new relationships and solution methods involving Somos sequences and Hankel transforms.

Contribution

It introduces a unified approach to solving Catalan-related recurrences using Riordan arrays and links these to elliptic curves and orthogonal polynomials, highlighting novel mathematical relationships.

Findings

Recurrences solved via Riordan arrays and Catalan numbers

Hankel transforms reveal Somos 4 sequences

Relations established between recurrences, elliptic curves, and orthogonal polynomials

Abstract

We show that the Catalan-Schroeder convolution recurrences and their higher order generalizations can be solved using Riordan arrays and the Catalan numbers. We investigate the Hankel transforms of many of the recurrence solutions, and indicate that Somos $4$ sequences often arise. We exhibit relations between recurrences, Riordan arrays, elliptic curves and Somos $4$ sequences. We furthermore indicate how one can associate a family of orthogonal polynomials to a point on an elliptic curve, whose moments are related to recurrence solutions.