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The main theme of this dissertation is retooling methods to work fordifferent situations.  I have taken the method derived by O'Hara and simplified by Zeilberger toprove unimodality of $q$-binomials and tweaked it. This allows us to createmany more families of polynomials for which unimodality is not, a priori,given. I analyze how many of the tweaks affect the resulting polynomial. Ayyerand Zeilberger proved a result about bounded lattice walks. I employ theirgenerating function relation technique to analyze lattice walks with a generalstep set in bounded, semi-bounded, and unbounded planes. The method in which wedo this is formulated to be highly algorithmic so that a computer can automatemost, if not all, of the work. I easily recover many well-known results forsimpler step sets and discover new results for more complex step sets.

Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics