Parameterized Telescoping Proves Algebraic Independence of Sums

TL;DR

This paper demonstrates that the absence of a parameterized telescoping solution can establish the algebraic independence of certain sums, providing new insights into summation theory and the limitations of Zeilberger's algorithm.

Contribution

It introduces a novel approach linking telescoping non-existence to algebraic independence and transcendence of sums, expanding the theoretical understanding of summation algorithms.

Findings

Non-existence of telescoping solutions implies algebraic independence.

This approach proves transcendence of specific classes of sums.

Provides explanation for limitations of Zeilberger's algorithm.

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.