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A Package on Formal Power Series

Wolfram Koepf
Organization: Universitšt Kassel
Department: AG Computational Mathematics
URL: http://www.mathematik.uni-kassel.de/~koepf/
Journal / Anthology

The Mathematica Journal
Year: 1994
Volume: 4
Issue: 2
Page range: 62-69

Formal Laurent-Puiseux series are important in many branches of mathematics. This paper presents SpecialFunctions.m (PowerSeries.m for Mathematica version 2.2), a Mathematica implementation of algorithms developed by the author for converting between certain classes of functions and their equivalent representing series. The package PowerSeries handles functions of rational, exponential and hypergeometric type, and enables the user to reproduce most of the results of Hansen's extensive table of series. Subalgorithms of independent significance generate differential equations satisfied by a given function and recurrence equations satisfied by a given sequence.

The SpecialFunctions and PowerSeries packagse can expand meromorphic functions of argument x^(1/p) with integer p of certain types into their corresponding Laurent-Puiseux series as a sum of expressions of the form

Sum[a[k](x-x0)^(m*k/p+s),{k,0,Infinity}] ,

where m is the 'symmetry number', s is the 'shift number', p is the 'Puiseux number' and x0 is the 'point of development'. The following types are supported:

functions of 'rational type', which are either rational or have a rational derivative of some order;

functions of 'hypergeometric type' where a[k+m]/a[k] is a rational function for some integer m;

functions of 'explike type' which satisfy a linear homogeneous differential equation with constant coefficients. These are the functions of the form Sum[p[k,x] Exp[a x],{k,1,n}] where p[k,x] are polynomials in x, and a is complex, or correspondingly sums with sines and cosines.

Further the package is able to convert this procedure, i.e. to calculate the generating function of a sequence.

A list of the Mathematica functions exported by PowerSeries is given by


and ?function yields a help message together with an example call.

The kernel function Series is extended to infinite series, try Series[E^x,{x,0}].

*Mathematics > Calculus and Analysis > Series

power series, Laurent series, Puiseux series, generating functions, ordinary differential equations, recurrence equations, hypergeometric functions, Pochhammer symbol, Bateman function, Hankel functions, Kummer functions, Whittaker functions, Struve functions, complimentary error functions, Abramowitz functions, parabolic cylinder functions
Related items

*Power Series and Generating Functions   [in MathSource: Packages and Programs]