Bryan Ek, Rutgers University

Abstract

The main goal of these projects was utilizing experimental mathematics to further our knowledge of several areas in math. We begin by tweaking a proof of unimodality by O'Hara to produce many more families of polynomials for which unimodality is not, a priori, given. I analyze how many of the tweaks affect the resulting polynomial. We then employ a generating function relation technique used by Ayyer and Zeilberger 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.