A Family of Bounded and Analytic Hyper-Operators

2021/05/28 by James David Nixon, Nixon, James David
Mathematics · #30B50 #30D05 #37F10 #39B12 #39B32 #Complex Variables (math.CV) #Differential Equations and Boundary Problems #Dynamical Systems (math.DS) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials #math.CV #math.DS #msc:30B50 #msc:30D05 #msc:37F10 #msc:39B12 #msc:39B32

paper · pdf · doi:10.48550/arxiv.2106.03935

arxiv created 2021/05/28 · openalex publication_date 2021/05/28 · arxiv updated 2021/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 α\uparrowⁿ z : (1,e^1/e) × ℂ_{\Re(z) > 0} → ℂ_{\Re(z) > 0} using methods from fractional calculus. This family satisfies the hyper-operator chain, α\uparrow^n-1 α\uparrowⁿ z = α\uparrowⁿ (z+1); with the initial condition α\uparrow⁰ z = α⋅ z.

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