Formulas for Continued Fractions. An Automated Guess and Prove Approach

TL;DR

The paper introduces an automated method to derive and prove formulas for continued fraction coefficients of special functions defined by differential equations, significantly streamlining the discovery process.

Contribution

It presents a novel automated approach combining conjecture generation and proof for continued fraction formulas based on differential equations.

Findings

Successfully generates closed-form formulas for continued fractions

Automates proof of formulas using linear recurrences

Captures a large part of existing literature on continued fractions

Abstract

We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial conditions. This is used to generate the first few coefficients and from there a conjectured formula. This formula is then proved automatically thanks to a linear recurrence satisfied by some remainder terms. Extensive experiments show that this simple approach and its straightforward generalization to difference and $q$-difference equations capture a large part of the formulas in the literature on continued fractions.