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

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Note: the submission process of this paper turned out to be very long (9 rounds!). Contrary to what is stated in the journal version, the first submission took place on 1998, March 19.