Computing Puiseux Expansions at Cusps of the Modular Curve X0(N)

The goal in this preprint is to give an efficient algorithm to compute Puiseux expansions at cusps of X0(N). It is based on a relation with a hypergeometric function that holds for any N.Comment: 4 page

The Cauchy-Kowalevski Theorem

We give a recursive description of polynomials with non-negative rational coefficients, which are coefficients of expansion in a power series solutions of partial differential equations in Cauchy-Kowalevski theorem

Fast computation of power series solutions of systems of differential equations

We propose new algorithms for the computation of the first N terms of a vector (resp. a basis) of power series solutions of a linear system of differential equations at an ordinary point, using a number of arithmetic operations which is quasi-linear with respect to N. Similar results are also given in the non-linear case. This extends previous results obtained by Brent and Kung for scalar differential equations of order one and two

Formulas for Continued Fractions. An Automated Guess and Prove Approach

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.Comment: Maple worksheet attache