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Fast Algorithms for Refined Parameterized Telescoping in Difference Fields

Carsten Schneider

TL;DR

This paper develops and analyzes fast algorithms for refined parameterized telescoping within the framework of $\Pi\Sigma^*$-fields, focusing on solving first-order parameterized linear difference equations (FPLDE) to obtain telescoping relations. The authors introduce a three-step strategy (denominator bound, degree bound, and degree reduction) to compute a complete solution space and then refine this to first-entry and reduced solutions, enabling telescoping and creative telescoping with minimal extension depth. A constructive version of Karr’s structural theorem is presented, enabling depth-optimal representations and efficient transformations of sum representations into reduced $\Pi\Sigma^*$-extensions. The methods are designed to handle indefinite nested sums/products (e.g., factorials, harmonic numbers) and are implemented in the Sigma package, with applications spanning combinatorics, numerics, number theory, and quantum field theory computations.

Abstract

Parameterized telescoping (including telescoping and creative telescoping) and refined versions of it play a central role in the research area of symbolic summation. Karr introduced 1981 $ΠΣ$-fields, a general class of difference fields, that enables one to consider this problem for indefinite nested sums and products covering as special cases, e.g., the ($q$--)hypergeometric case and their mixed versions. This survey article presents the available algorithms in the framework of $ΠΣ$-extensions and elaborates new results concerning efficiency.

Fast Algorithms for Refined Parameterized Telescoping in Difference Fields

TL;DR

This paper develops and analyzes fast algorithms for refined parameterized telescoping within the framework of -fields, focusing on solving first-order parameterized linear difference equations (FPLDE) to obtain telescoping relations. The authors introduce a three-step strategy (denominator bound, degree bound, and degree reduction) to compute a complete solution space and then refine this to first-entry and reduced solutions, enabling telescoping and creative telescoping with minimal extension depth. A constructive version of Karr’s structural theorem is presented, enabling depth-optimal representations and efficient transformations of sum representations into reduced -extensions. The methods are designed to handle indefinite nested sums/products (e.g., factorials, harmonic numbers) and are implemented in the Sigma package, with applications spanning combinatorics, numerics, number theory, and quantum field theory computations.

Abstract

Parameterized telescoping (including telescoping and creative telescoping) and refined versions of it play a central role in the research area of symbolic summation. Karr introduced 1981 -fields, a general class of difference fields, that enables one to consider this problem for indefinite nested sums and products covering as special cases, e.g., the (--)hypergeometric case and their mixed versions. This survey article presents the available algorithms in the framework of -extensions and elaborates new results concerning efficiency.

Paper Structure

This paper contains 10 sections, 17 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. A short summary of $\Pi\Sigma^*$-field theory and $\Pi\Sigma^*$-extensions
  3. Solving parameterized first-order equations
  4. A general strategy
  5. Turning the strategy to algorithms
  6. Special case: parameterized telescoping
  7. Parameterized first-entry telescoping solutions
  8. Refined parameterized telescoping: reduced solutions
  9. A constructive version of Karr's structural theorem
  10. Conclusion

Key Result

theorem 1

[Karr:81Karr:85] Consider the difference field extension $({\mathbb{F}(t)},{\sigma})$ of $({\mathbb{F}},{\sigma})$ with $t$ being transcendental over $\mathbb{F}$ and $\sigma(t)=t+\beta$. Then this is a $\Sigma^*$-extension iff there is no $g\in\mathbb{F}$ with $\sigma(g)=g+\beta$.

Theorems & Definitions (36)

  • definition 1
  • definition 2
  • theorem 1
  • proposition 1
  • definition 3
  • theorem 2
  • definition 4
  • remark 1
  • definition 5
  • theorem 3: Schneider:06d
  • ...and 26 more