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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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A difference ring theory for symbolic summation.

Carsten Schneider1

  • 1Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Altenbergerstraße 69, 4040 Linz, Austria.

Journal of Symbolic Computation
|January 5, 2016
PubMed
Summary
This summary is machine-generated.

A new summation framework enhances symbolic computation by handling complex nested sums and products. This approach provides algorithms for solving difference equations, aiding in fields like combinatorics and particle physics.

Keywords:
Difference ring extensionsIndefinite nested sums and productsParameterized telescoping (telescoping creative telescoping)Roots of unitySemi-constantsSemi-invariants

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Area of Science:

  • Computer Science
  • Mathematics
  • Symbolic Computation

Background:

  • Karr's difference field approach is a foundational method for symbolic summation.
  • Existing methods face limitations with certain types of nested products, such as those over roots of unity.

Purpose of the Study:

  • To develop an enhanced summation framework building upon Karr's approach.
  • To extend the capabilities to include transcendental extensions and products over roots of unity.
  • To provide algorithms for automatic construction of difference rings and solving difference equations.

Main Methods:

  • Development of a summation framework based on difference ring theory.
  • Implementation of algorithms for constructing difference rings.
  • Algorithms for solving parameterized telescoping and first-order difference equations.

Main Results:

  • The framework successfully handles indefinite nested sums and products, including those over roots of unity.
  • Algorithms for automatic difference ring construction are presented.
  • Efficient algorithms for solving parameterized difference equations are derived.

Conclusions:

  • The developed summation machinery offers a rigorous approach to symbolic summation.
  • The framework has been successfully applied to complex problems in combinatorics and particle physics.
  • This work advances the field of symbolic computation for challenging mathematical problems.