Homogeneous Linear Calculus of Order 1 and a $\lambda$-Taylor Formula

TL;DR

This paper introduces a novel calculus on sequences involving a $\lambda$-derivative and $\lambda$-integral, establishing fundamental theorems, function bases, and a $\lambda$-Taylor formula for advanced sequence analysis.

Contribution

It develops a new sequence calculus with $\lambda$-derivatives and integrals, including fundamental theorems and a $\lambda$-Taylor expansion, advancing mathematical tools for sequence analysis.

Findings

Established the fundamental theorem of $\lambda$-calculus.

Constructed a suitable function basis for $\lambda$-derivative and integral.

Derived a $\lambda$-Taylor formula for functions.

Abstract

In this paper, a new calculus on sequences is defined. Also, the $\lambda$-derivative and the $\lambda$-integration are investigated. The fundamental theorem of $\lambda$-calculus is included. A suitable function basis for the $\lambda$-derivative and the $\lambda$-integral is provided, and various properties of this basis are given. A $\lambda$-Taylor formula for functions is given.