Continued Fraction as a Discrete Nonlinear Transform

TL;DR

This paper explores how continued fractions serve as a nonlinear transform that simplifies complex sequences, such as those arising in graph combinatorics, by relating Taylor series coefficients to continued-fraction coefficients.

Contribution

It demonstrates the nonlinear relation between Taylor series and continued fractions and shows how this transform simplifies sequences in graph combinatorics.

Findings

Continued fractions can transform complicated sequences into simpler ones.

The nonlinear relation connects Taylor coefficients with continued-fraction coefficients.

Application to graph combinatorics illustrates the simplification process.

Abstract

The connection between a Taylor series and a continued-fraction involves a nonlinear relation between the Taylor coefficients $\{ a_n \}$ and the continued-fraction coefficients $\{ b_n \}$. In many instances it turns out that this nonlinear relation transforms a complicated sequence $\{a_n \}$ into a very simple one $\{ b_n \}$. We illustrate this simplification in the context of graph combinatorics.