High-order differential approximants

We introduce a new form of differential approximant for the summation of power series. The method is a special type of Padé–Hermite approximant. It consists of a high-order linear differential equation with polynomial coefficients that is satisfied approximately by the partial sum of the power series. This method is able to reproduce the polylogarithmic functions exactly. Numerical evidence suggests that this is currently one of the best methods of singularity analysis for many problems.