A Telescoping method for Double Summations

We present a method to prove hypergeometric double summation identities. Given a hypergeometric term $F(n,i,j)$, we aim to find a difference operator $L=a_0(n) N^0 + a_1(n) N^1 +...+a_r(n) N^r$ and rational functions $R_1(n,i,j),R_2(n,i,j)$ such that $L F = \Delta_i (R_1 F) + \Delta_j (R_2 F)$. Based on simple divisibility considerations, we show that the denominators of $R_1$ and $R_2$ must possess certain factors which can be computed from $F(n, i,j)$. Using these factors as estimates, we may find the numerators of $R_1$ and $R_2$ by guessing the upper bounds of the degrees and solving systems of linear equations. Our method is valid for the Andrews-Paule identity, Carlitz's identities, the Ap\'ery-Schmidt-Strehl identity, the Graham-Knuth-Patashnik identity, and the Petkov\v{s}ek-Wilf-Zeilberger identity.Comment: 22 pages. to appear in J. Computational and Applied Mathematic

Bounds for D-finite closure properties

We provide bounds on the size of operators obtained by algorithms for executing D-finite closure properties. For operators of small order, we give bounds on the degree and on the height (bit-size). For higher order operators, we give degree bounds that are parameterized with respect to the order and reflect the phenomenon that higher order operators may have lower degrees (order-degree curves)

Computer algebra tools for Feynman integrals and related multi-sums

In perturbative calculations, e.g., in the setting of Quantum Chromodynamics (QCD) one aims at the evaluation of Feynman integrals. Here one is often faced with the problem to simplify multiple nested integrals or sums to expressions in terms of indefinite nested integrals or sums. Furthermore, one seeks for solutions of coupled systems of linear differential equations, that can be represented in terms of indefinite nested sums (or integrals). In this article we elaborate the main tools and the corresponding packages, that we have developed and intensively used within the last 10 years in the course of our QCD-calculations

Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations

We outline a new algorithm to solve coupled systems of differential equations in one continuous variable $x$ (resp. coupled difference equations in one discrete variable $N$) depending on a small parameter $\epsilon$: given such a system and given sufficiently many initial values, we can determine the first coefficients of the Laurent-series solutions in $\epsilon$ if they are expressible in terms of indefinite nested sums and products. This systematic approach is based on symbolic summation algorithms in the context of difference rings/fields and uncoupling algorithms. The proposed method gives rise to new interesting applications in connection with integration by parts (IBP) methods. As an illustrative example, we will demonstrate how one can calculate the $\epsilon$-expansion of a ladder graph with 6 massive fermion lines