Formal power series

In this article we will describe the \Maple\ implementation of an algorithm presented in~\cite{Koe92}--\cite{Koeortho} which computes an {\em exact\/} formal power series (FPS) of a given function. This procedure will enable the user to reproduce most of the results of the extensive bibliography on series~\cite{Han}. We will give an overview of the algorithm and then present some parts of it in more detail

Refined Holonomic Summation Algorithms in Particle Physics

An improved multi-summation approach is introduced and discussed that enables one to simultaneously handle indefinite nested sums and products in the setting of difference rings and holonomic sequences. Relevant mathematics is reviewed and the underlying advanced difference ring machinery is elaborated upon. The flexibility of this new toolbox contributed substantially to evaluating complicated multi-sums coming from particle physics. Illustrative examples of the functionality of the new software package RhoSum are given.Comment: Modified Proposition 2.1 and Corollary 2.

Explicit formula for the generating series of diagonal 3D rook paths

Let $a_n$ denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an $n \times n \times n$ three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computer-driven \emph{discovery and proof} of the fact that the generating series $G(x)= \sum_{n \geq 0} a_n x^n$ admits the following explicit expression in terms of a Gaussian hypergeometric function: G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.Comment: To appear in "S\'eminaire Lotharingien de Combinatoire