On The Indefinite Sum In Fractional Calculus

TL;DR

This paper develops a new framework for indefinite summation of complex functions using fractional calculus transforms, enabling summation of more complex functions and introducing a convolution-based operator with properties akin to fractional derivatives.

Contribution

It introduces a novel indefinite summation operator for holomorphic functions via fractional calculus, with a unique definition, convolution structure, and complex iterates resembling Riemann-Liouville differintegrals.

Findings

Established a theorem for indefinite summation of holomorphic functions with exponential bounds.

Defined a convolution operation for the indefinite sum that is commutative, associative, and distributive.

Derived a formula for complex iterates of the indefinite sum similar to fractional differintegrals.

Abstract

We present a theorem on taking the repeated indefinite summation of a holomorphic function $\phi(z)$ in a vertical strip of $\mathbb{C}$ satisfying exponential bounds as the imaginary part grows. We arrive at this result using transforms from fractional calculus. This affords us the ability to indefinitely sum more complicated functions than previously possible; such as holomorphic functions of order $n \in \mathbb{N}$ that have decay at plus or minus imaginary infinity. We then further investigate the indefinite summation operator by restricting ourselves to a space of functions of exponential type. We arrive at a second representation for the indefinite summation operator, equivalent to the first presented, and show we have defined a unique operator on this space. We develop a convolution using the indefinite sum that is commutative, associative, and distributive over addition. We arrive at a formula for the complex iterates of the indefinite sum (the differsum), using this convolution, that resembles the Riemann-Liouville differintegral. We close with a generalization of the Gamma function.