Many multi-sums and integrals coming from challenging problems in mathematics (e.g., number theory, combinatorics, analysis, numerics,...), computer science (e.g., analysis of algorithms, proof verification, ...) or natural sciences (e.g. elementary particle phsyics) can be simplified by symbolic methods to closed forms. This means to expressions in terms of special functions that are easier to handle and/or that provide extra information on the given input sum or integral. In this lecture we will elaborate on crucial symbolic summation and integration methods, often available in major computer algebra systems, that work in the setting of difference rings (for sums) and differential rings (for integrals).

Here we will start with the underlying computer algebra methods for sums initiated by Michael Karr that have been improved and generalized further to a general difference ring theory for big classes of multi-sums in terms of indefinite nested sums and products. The underlying methods enable one to solve the telescoping problem for indefinite summation, parametericed telescoping for finding linear recurrences or differential equations of sums depending on an extra parameter, or solving linear recurrences in terms of certain classes of special functions. The interaction of all these algorithms will lead to a general framework to tackle challenging multi-sums that arise in various disciplines of technical and natural sciences that have been indicated above.

In the second part of the lecture we will present the continuous analogue of symbolic summation that has been initiated by Risch's algorithm. This algrorithm decides if there is a solution in terms of elementary functions of the inverse problem of differentiation (also called anti-differentiation, the continuous version of telescoping). In particular, we will elaborate the similarities but also the differences of the available integration methods in comparison to the existing symbolic summation techniques. Here we will consider also the parameterized anti-differentiation problem to find linear differential equations or recurrences of integrals depending on an extra parameter and solving linear differential equations in terms of special functions.

Summarizing, this lectures aims at introducing the crucial algorithms for symbolic summation and integration and providing the basic toolboxes to tackle non-trivial problems that arise in technical and natural sciences.

Examples in Mathematica complementing the lecture can be found here. Supplementing slides are available here.

The lecture is based on ideas that arise in parts in

• S.A. Abramov, M. Bronstein, M. Petkovsek, C. Schneider: On Rational and Hypergeometric Solutions of Linear Ordinary Difference Equations in ΠΣ-field extensions. J. Symb. Comput. 107, pp. 23-66. arXiv:2005.04944 [cs.SC].
• M. Bronstein: Symbolic Integration I, Transcendental Functions
• K.O. Geddes, S.R. Czapor, G. Labahn: Algorithms for Computer Algebra
• C. Schneider Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation. In: Anti-Differentiation and the Calculation of Feynman Amplitudes, pp. 423-485. 2021. Springer, arXiv:2102.01471 [cs.SC]
• C. Schneider: A Difference Ring Theory for Symbolic Summation. J. Symb. Comput. 72, pp. 82-127. 2016. arXiv:1408.2776 [cs.SC]
• C. Schneider: Summation Theory II: Characterizations of RΠΣ-extensions and algorithmic aspects. J. Symb. Comput. 80(3), pp. 616-664. 2017. arXiv:1603.04285 [cs.SC]