## Mathematics, Department of

#### First Advisor

Allan Peterson

#### Date of this Version

8-2018

#### Abstract

In this dissertation we develop a fractional difference calculus for functions on a discrete domain. We start by showing that the Taylor monomials, which play a role analagous to that of the power functions in ordinary differential calculus, can be expressed in terms of a family of polynomials which I will refer to as the Pochhammer polynomials. These important functions, the Taylor monomials, were previously described by other scholars primarily in terms of the gamma function. With only this description it is challenging to understand their properties. Describing the Taylor monomials in terms of the Pochhammer polynomials has made it easier to understand their behavior, as we demonstrate in this work. We then use the Taylor monomials to define a fractional operator which generalizes the standard (iterated) backward difference operator. We show that these fractional difference operators have a very simple composition rule and act nicely on the Taylor monomials. We then describe the Riemann-Liouville and Caputo fractional difference operators in terms of this more general fractional operator and use the properties of the general fractional operator to derive the composition rules for all such operators. Finally, we apply this theory to study a nonlinear boundary value problem described using a Caputo fractional difference and show how to obtain a sequence of approximate solutions which converges quadratically to the unique solution to this problem.

Adviser: Allan Peterson

## Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Allan Peterson. Lincoln, Nebraska : August, 2018

Copyright (c) 2018 Scott C. Gensler