A package on formal power series

Formal Laurent-Puiseux series of the form f(x) = ∑ k k0 akx k/n (1) are important in many branches of mathematics. Whereas Mathematica supports the calculation of truncated series with its Series command, and the Mathematica package SymbolicSum that is shipped with Mathematica version 2 is able to convert formal series of type (1) in some instances to their corresponding generating functions, in [2]–[7] we developed an algorithmic procedure to do these conversions that is implemented by the author, A. Rennoch and G. Stölting in the Mathematica package PowerSeries. The implementation enables the user to reproduce most of the results of the extensive bibliography on series [Hansen [1], 1975]. Moreover a subalgorithm of its own significance generates differential equations satisfied by the input function. Scope of the algorithm In [2]–[4] three types of functions are covered by an algorithmic procedure for the conversion into their representing Laurent-Puiseux series (1) at the origin: functions of rational type which are rational, or have a rational derivative of some order, functions of exp-like type which satisfy a homogeneous linear differential equation with constant coefficients, and functions of hypergeometric type which have a representation (1) with coefficients satisfying a recurrence equation of the form ak+m = R(k) ak for k ≥ k0 ak = Ak for k = k0, k0 + 1, . . . , k0 + m− 1 for some m ∈ IN, Ak ∈ C (k = k0+1, k0+2, . . . , k0+m−1), Ak0 ∈ C\{0}, and some rational function R. The number m is then called the symmetry number of (the given representation of) f . The most interesting case is formed by the functions of hypergeometric type as almost all transcendental elementary functions like x^n, E^x, Log[x], Sin[x], Cos[x], ArcSin[x],