Fast evaluation of holonomic functions near and in regular singularities
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Abstract

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 abnormally long: 9 rounds! Contrary to what is stated on the journal version, the first submission took place on 1998, March 19.