In a series of previous articles, we have given efficient algorithms for the evaluation of holonomic functions over the algebraic numbers and for the computation of their limits at singularities. The focus of these articles was mainly on the efficient evaluation at a fixed point. In the present note, we will show that there exist uniformly efficient algorithms for evaluating holonomic functions. The main technical difficulty is to maintain uniform efficiency near irregular singularities. We will introduce a variant of accelerato-summation for this purpose that we call “expedito-summation”.

Keywords: holonomic function, special function, fast evaluation, accelero-summation, expedito-summation

A.M.S. subject classification: 33-04, 30-04, 40-04, 33F05, 33E30, 40G10, 30B40

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