Riordan arrays and Jacobi and Thron continued fractions

TL;DR

This paper explores the connection between Riordan arrays and continued fractions of Jacobi and Thron types, revealing new links to orthogonal polynomials and transformations within the Riordan group.

Contribution

It demonstrates how generating functions of certain Riordan arrays can be expressed as continued fractions and investigates their inverses, linking to orthogonal polynomial sequences and Riordan group involutions.

Findings

Riordan arrays have generating functions as Jacobi and Thron continued fractions.

Inverses of these arrays can also be expressed via continued fractions.

Transformations of continued fractions lead to exponential Riordan arrays.

Abstract

We show that certain Riordan arrays have generating functions that can be expressed as continued fractions of Jacobi and Thron type. We investigate the inverses of such arrays, which in certain circumstances can also have generating functions representable as continued fractions. Links to orthogonal polynomial moment sequences, and to Laurent biorthogonal polynomials are developed. We show that certain Riordan group involutions can be defined by continued fractions. We also show how simple transformations of the Jacobi continued fractions can lead to exponential Riordan arrays. Finally, by way of contrast, we look at the case of some non Riordan arrays that are of combinatorial significance, including the Narayana numbers.