Towards a symbolic summation theory for unspecified sequences

TL;DR

This paper develops a symbolic summation framework for simplifying indefinite double sums involving unspecified sequences, identifying when such sums can be reduced to single sums or require nested sums with specific constraints.

Contribution

It introduces a summation machinery that determines when indefinite double sums can be simplified to single sums or nested sums with constraints, especially for hypergeometric and related sequences.

Findings

The machinery can simplify certain double sums without exceptions.

When simplification fails, a nested sum with a specific constraint is introduced.

The constraint is both necessary and sufficient for hypergeometric and related sequences.

Abstract

The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide such a simplification without exceptions. If it fails, it may suggest a more advanced simplification introducing in addition a single nested sum where the summand has to satisfy a particular constraint. More precisely, an explicitly given parameterized telescoping equation must hold. Restricting to the case that the arising unspecified sequences are specialized to the class of indefinite nested sums defined over hypergeometric, multi-basic or mixed hypergeometric products, it can be shown that this constraint is not only sufficient but also necessary.