Open Access

Wolfram Koepf

• In many applications (hypergeometric-type) special functions like orthogonal polynomials are needed. For example in more than 50 \% of the published solutions for the (application-oriented) questions in the Problems Section'' of SIAM Review special functions occur. In this article the Mathematica package {\tt SpecialFunctions} which can be obtained from the URL {\tt http://www.zib.de/koepf} is introduced. Algorithms to convert between power series representations and their generating functions is the main topic of this package, extending the previous package {\tt PowerSeries}. Moreover the package automatically finds differential and recurrence equations for expressions and for sums (the latter using Zeilberger's algorithm. As an application the fast computation of polynomial approximations of solutions of linear differential equations with polynomial coefficients is presented. This is the asymptotically fastest known algorithm for series computations, and it is much faster than Mathematica's builtin {\tt Series} command if applicable. Many more applications are considered. Finally the package includes implementations supporting the efficient computation of classical continuous and discrete orthogonal polynomials.