








Power Series and Generating Functions






Organization:  Universität Kassel 
Department:  AG Computational Mathematics 






0207379






19980413






This package can expand meromorphic functions of argument x^(1/p) with integer p of certain types into their corresponding LaurentPuiseux series as a sum of expressions of the form Sum[a[k](xx0)^(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, 6269" 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}].












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












 SpecialFunctions.m (264.1 KB)  Mathematica Package   SpecialFunctions.ps (215.4 KB)  PostScript Documentation  Files specific to Mathematica 2.2 version:
  PowerSeries.txt (3.3 KB)  Test file   PowerSeries.m (134 KB)  Mathematica Package 







   
 
