Discrete Calculus of Finite Sequences

TL;DR

This paper extends the calculus of finite differences to finite sequences, enabling the definition of convexity and related concepts, and proposes a method to connect this extension back to the classical calculus.

Contribution

It introduces a novel extension of finite difference calculus to finite sequences and establishes a link to traditional calculus methods.

Findings

Extended calculus for finite sequences is formulated.

Convexity for finite sequences is defined.

A method to connect the extension to classical calculus is proposed.

Abstract

The calculus of finite differences is a solid foundation for the development of operations such as the derivative and the integral for infinite sequences. Here we showed a way to extend it for finite sequences. We could then define convexity for finite sequences and some related concepts. To finalize, we propose a way to go from our extension to the calculus of finite differences.