Telescopic, Multiplicative, and Rational Extensions of Summations

TL;DR

This paper explores advanced methods for extending summations, including telescoping, multiplicative closures, and rational extensions, to broaden the applicability of summation techniques in algebraic and analytical contexts.

Contribution

It introduces and analyzes rational extensions of summations, generalizing telescoping and studying multiplicative closures that are not inherently closed.

Findings

Develops a framework for rational extensions of summations.

Analyzes multiplicative closures that are not multiplicatively closed.

Provides new insights into extending summations beyond traditional methods.

Abstract

A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we revisit telescoping, we study multiplicative closures of summations (such as the usual summation on convergent series) that are not themselves multiplicatively closed, and we study rational extensions as a generalization of telescoping.