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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