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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.