Assoc. Prof. Dr. Carsten Schneider
RISC - Research Institute for Symbolic Computation
Johannes Kepler University Linz
Carsten.Schneider@risc.jku.at (Tel: +43 732 2468 9966)

Overview

I and my research group Computer algebra for Quantum Field Theory, closely connected with the RISC combinatorics group, develop computer algebra algorithms for nested sums and products with the task to apply them, e.g., to combinatorial problems, number theory, numerics, and particle physics.

Among others, I focus on the following two aspects:

Multi-summation in difference rings

Within difference fields and rings I am developing summation theories that enable one to simplify definite nested multi-sums to representations in terms of indefinite nested sums and products. Special emphasis is put on representations which are optimal, e.g., concerning their nested depth. The resulting algorithms of the underlying summation theory are implemented in the summation package Sigma.
Among other features, the following tasks can be handled:

  • Compute recurrences (based on the paradigm of Z's creative telescoping) for definite sums with summands given in terms of indefinite nested sums and products.
  • Solve recurrences in terms of all solutions that are expressible in terms of indefinite nested sums and products (d'Alembertian solutions).
  • Eliminate all algebraic relations among the summation objects and simplify them to representations with optimal nesting depth.

All three components combined deliver strong tools in order to compute closed forms of summation problems. For more details see, e.g.

  • C. Schneider. Symbolic Summation Assists Combinatorics.Sem. Lothar. Combin.56, pp. 1-36.2007..Article B56b. [url] [ps] [pdf] [bib]
  • C. Schneider. Simplifying Multiple Sums in Difference Fields. In: Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, J. Bl├╝mlein, C. Schneider (ed.), Texts and Monographs in Symbolic Computation. 2013. Springer. [url] [bib]
  • C. Schneider. Modern Summation Methods for Loop Integrals in Quantum Field Theory: The Packages Sigma, EvaluateMultiSums and SumProduction. In: Proc. ACAT 2013, J. Phys.: Conf. Ser.523/012037, pp. 1-17.2014.[url] [bib]

Exploring nested sums and products

In order to simplify multi-sums (with infinite summation ranges) to indefinite nested sums and products, algebraic and analytic properties of indefinite nested sums are crucial. In particular, I am interested, e.g., in the following problems:

  • Prove algebraic independence of indefinite nested sums and products exploiting difference ring theory.
  • Find algebraic relations of infinite nested sums (multiple zeta values and generalizations of them) using the so far understood algebraic relations induced, e.g., by shuffle algebras, stuffle algebras, duplication relations, etc.
  • Compute asymptotic expansions of indefinite nested sums by exploiting, e.g., integral representations (Mellin transform).