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Parameterized Telescoping Proves Algebraic Independence of Sums

Carsten Schneider

TL;DR

The paper shows that the nonexistence of parameterized telescoping solutions in a $\Pi\Sigma^*$-extension framework implies algebraic independence of associated sums, enabling automatic transcendence results and shedding light on recurrence minimality in Zeilberger-style algorithms. By constructing a ring monomorphism from generalized d'Alembertian extensions to the ring of sequences, it transfers transcendence properties to sequences of partial sums and products. The authors develop a concrete decision criterion for algebraic independence, apply it to rational, hypergeometric, and nested sums, and extend the framework to products, with broad algorithmic consequences via the Sigma package. These results unify telescoping, transcendence theory, and symbolic summation, predicting independence for wide classes of sums and clarifying when recurrences of minimal order exist. The work has practical implications for automated summation and for understanding the limits of creative telescoping in deriving compact recurrences.

Abstract

Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic independence of certain types of sums. Combining this fact with summation-theory shows transcendence of whole classes of sums. Moreover, this result throws new light on the question why, e.g., Zeilberger's algorithm fails to find a recurrence with minimal order.

Parameterized Telescoping Proves Algebraic Independence of Sums

TL;DR

The paper shows that the nonexistence of parameterized telescoping solutions in a -extension framework implies algebraic independence of associated sums, enabling automatic transcendence results and shedding light on recurrence minimality in Zeilberger-style algorithms. By constructing a ring monomorphism from generalized d'Alembertian extensions to the ring of sequences, it transfers transcendence properties to sequences of partial sums and products. The authors develop a concrete decision criterion for algebraic independence, apply it to rational, hypergeometric, and nested sums, and extend the framework to products, with broad algorithmic consequences via the Sigma package. These results unify telescoping, transcendence theory, and symbolic summation, predicting independence for wide classes of sums and clarifying when recurrences of minimal order exist. The work has practical implications for automated summation and for understanding the limits of creative telescoping in deriving compact recurrences.

Abstract

Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic independence of certain types of sums. Combining this fact with summation-theory shows transcendence of whole classes of sums. Moreover, this result throws new light on the question why, e.g., Zeilberger's algorithm fails to find a recurrence with minimal order.

Paper Structure

This paper contains 10 sections, 26 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Basic notions: $\Pi\Sigma^*$-extensions and generalized d'Alembertian extension
  3. Parameterized telescoping, $\Pi\Sigma^*$-extensions and the ring of sequences
  4. The monomorphism construction
  5. A criterion to check algebraic independence
  6. Rational sums
  7. Hypergeometric sums and the minimality of recurrences
  8. Nested sums
  9. A transcendence criterion for products
  10. Conclusion

Key Result

Theorem 2.3

Let $({\mathbb{F}(t)},{\sigma})$ be a difference field extension of $({\mathbb{F}},{\sigma})$ with $\sigma(t)=\alpha\,t+\beta$ where $\alpha\in\mathbb{F}^*$ and $\beta\in\mathbb{F}$. Then:

Theorems & Definitions (52)

  • Definition 2.1
  • Example 2.2
  • Theorem 2.3: Karr:81
  • Theorem 2.4
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7
  • proof
  • Theorem 2.8
  • Corollary 2.9
  • ...and 42 more