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Power Series and Generating Functions
Author

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

0207-379
Revision date

1998-04-13
Description

This package 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.

The package is described in detail in "Wolfram Koepf: A package on formal power series, The Mathematica Journal 4, 1994, 62-69"

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

?PowerSeries`*

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}].
Subjects

*Mathematics > Calculus and Analysis > Series
*Mathematics > Number Theory
Keywords

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

*A Package on Formal Power Series   [in Articles]
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Files specific to Mathematica 2.2 version:
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PowerSeries.txt (3.3 KB) - Test file
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PowerSeries.m (134 KB) - Mathematica Package