A Family of Bounded and Analytic Hyper-Operators

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 $\alpha {\uparrow }^{0}z=\alpha \cdot z$.

References in corpus (1)

• On The Indefinite Sum In Fractional Calculus