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.
Sébastien Maulat, Bruno Salvy. Formulas for Continued Fractions. An Automated Guess and Prove Approach. ISSAC'15, Jul 2015, Bath, United Kingdom. ⟨10.1145/2755996.2756660⟩. ⟨hal-01227259⟩