ISSN:
1432-0622
Keywords:
Keywords: formal power series
;
Laurent-Puiseux series
;
closed forms
;
hypergeometric terms and functions
;
functions of hypergeometric type
;
holonomic linear differential and recurrence equations.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
,
Technology
Notes:
Abstract. There are several well-known algorithms to calculate the Puiseux series developments of the branches of an algebraic function. None of them, however, generates the series in closed form, even in those cases where such a formal result is available. They produce, instead, truncated series, and give information that can be used to handle the series as streams. Here we give a solution to the given problem. We combine an algorithm of D. V. and G. V. Chudnovsky that transforms the given algebraic equation into a differential equation for the function, and further into a recurrence equation for the Puiseux coefficients, with an algorithm of Koepfwhich in the case of hypergemetric type results in the formal series. A finitelinear recurrence equation is optimal for a representation by streams. D. V. and G. V. Chudnovsky point out that their algorithm requires only O (M) fieldoperations if M is the order of the number of series terms considered. However,from a practical point of view, it is of importance that the complexity of the resulting recurrence equation – as well as of the differential equation – can be extremelyhigh, a fact, which we illustrate by an example. It turns out, that many alge-braic functions of low order with a sparse representation are of hypergeometrictype, and so closed form representations for the corresponding series can be given.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01613613
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