A Family of Bounded and Analytic Hyper-Operators

TL;DR

This paper constructs a family of analytic hyper-operators using fractional calculus, extending the classical hyper-operator chain with new bounded and analytic functions.

Contribution

It introduces a novel family of hyper-operators that are bounded, analytic, and satisfy the hyper-operator chain, expanding the mathematical understanding of hyper-operator functions.

Findings

Defines a family of analytic hyper-operators on a specific domain.

Establishes the hyper-operator chain property for the family.

Provides a framework using fractional calculus methods.

Abstract

This is a summation of research done in the author's second and third year of undergraduate mathematics at The University of Toronto. As the previous details were largely scattered and disorganized; the author decided to rewrite the cumulative research. The goal of this paper is to construct a family of analytic functions $\alpha \uparrow^n z : (1,e^{1/e}) \times \mathbb{C}_{\Re(z) > 0} \to \mathbb{C}_{\Re(z) > 0}$ using methods from fractional calculus. This family satisfies the hyper-operator chain, $\alpha \uparrow^{n-1} \alpha \uparrow^n z = \alpha \uparrow^n (z+1)$; with the initial condition $\alpha \uparrow^0 z = \alpha \cdot z$.