By Bryan Ek

First Written: March 29th, 2018; This version: April 16th, 2018.

Submitted to Rutgers University to fulfill the dissertation requirement of a doctoral degree.

Abstract:
The main theme of this dissertation is retooling methods to work for different situations.

I have taken the method derived by O'Hara and simplified by Zeilberger to prove unimodality of $q$-binomials and tweaked it. This allows us to create many more families of polynomials for which unimodality is not, a priori, given. I analyze how many of the tweaks affect the resulting polynomial.

Ayyer and Zeilberger proved a result about bounded lattice walks. I employ their generating function relation technique to analyze lattice walks with a general step set in bounded, semi-bounded, and unbounded planes. The method in which we do this is formulated to be highly algorithmic so that a computer can automate most, if not all, of the work. I easily recover many well-known results for simpler step sets and discover new results for more complex step sets.

- Gnk.txt, for discovery of symmetric and unimodal polynomials.
- ScoringPaths.txt, for discovering generating functions of bounded, semi-bounded, and unbounded scoring paths.

Articles of Bryan Ek