Series reversion with Jacobi and Thron continued fractions

TL;DR

This paper investigates the reversion of power series defined by second-order recurrences using generating functions, expressing results through Jacobi and Thron continued fractions and relating them to Eulerian expressions.

Contribution

It introduces a novel approach to express series reversion via Jacobi and Thron continued fractions and connects these with Eulerian transformations.

Findings

Reversion formulas expressed with Jacobi and Thron continued fractions

Relations established between continued fractions and Eulerian expressions

Analytical framework for second-order recurrence series reversion

Abstract

Using ordinary and exponential generating functions, we explore the reversion of power series defined by $2$nd order recurrences. We express the reversions in terms of Jacobi and Thron continued fractions. We find relations with Eulerian expressions using a transformation of continued fractions.