Power Series Representations of Hypergeometric Type Functions

Abstract

AbstractIn 1992, Koepf proposed a symbolic approach to compute power series. This algorithm was extended for a larger family of expressions thanks to Petkovsek’s and van Hoeij’s algorithms (1993 and 1998) which compute hypergeometric term solutions of any given holonomic recurrence equation (RE). Mark van Hoeij’s algorithm whose outputs are bases is available in Maple through the command LREtools[hypergeomsols], and Koepf’s algorithm through convert and the built-in module FormalPowerSeries. LREtools[hypergeomsols] is internally used by convert/FormalPowerSeries.However, using van Hoeij’s algorithm one cannot compute m-fold hypergeometric term solutions of holonomic REs, for integers \(m>1\). Given a field K of characteristic zero, a term a(n) is said to be m-fold hypergeometric if the term ratio \(a(n+m)/a(n)\) is rational over K. Note that the hypergeometric term case corresponds to \(m=1\). If one adds for example an odd hypergeometric function, like \(\arcsin (z)\), and an even hypergeometric function, like \(\cos (z)\) (which both are two-fold hypergeometric), then van Hoeij’s algorithm cannot find those by solving the resulting recurrence equation. Due to this limitation, the computation of many power series is missed by Maple, in particular, linear combinations of power series having m-fold hypergeometric term coefficients are generally not detected.We overcome these issues by using a new algorithm called mfoldHyper, proposed in the first author’s Ph.D. thesis to compute bases of the subspace of m-fold hypergeometric term solutions of holonomic REs. It turns out that mfoldHyper linearizes the computation of hypergeometric type power series, i.e. every linear combination of hypergeometric type power series is detected. This paper describes our Maple implementation of an algorithm that conclusively extends Maple’s capabilities regarding the computation of hypergeometric type power series.KeywordsHypergeometric type power seriesm-fold hypergeometric termHolonomic recurrence equation