Fast Evaluation of Holonomic Functions Near and in Regular Singularities

A holonomic function is an analytic function, which satisfies a linear differential equationLf = 0 with polynomial coefficients. In particular, the elementary functions exp,log,sin, etc., and many special functions such as erf, Si, Bessel functions, etc., are holonomic functions. In a previous paper, we have given an asymptotically fast algorithm to evaluate a holonomic function f at a non-singular point z′ on the Riemann surface of f, up to any number of decimal digits while estimating the error. However, this algorithm becomes inefficient, when z′ approaches a singularity of f. In this paper, we obtain efficient algorithms for the evaluation of holonomic functions near and in singular points where the differential operator L is regular (or, slightly more generally, where L is quasi-regular—a concept to be introduced below).