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The purpose of this dissertation is to develop and apply results of both discrete calculus and discrete fractional calculus to further develop results on various discrete time scales. Two main goals of discrete and fractional discrete calculus are to extend results from traditional calculus and to unify results on the real line with those on a variety of subsets of the real line. Of particular interest is introducing and analyzing results related to a generalized fractional boundary value problem with Lidstone boundary conditions on a standard discrete domain N_a. We also introduce new results regarding exponential order for functions on quantum time scales, along with extending previously discovered results. Finally, we conclude by introducing and analyzing a boundary value problem, again with Lidstone boundary conditions, on a mixed time scale, which may be thought of as a generalization of the other time scales in this work.
Advisers: Lynn Erbe and Allan Peterson