Date of Award

11-2015

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematical Sciences

First Advisor

Dr. Mohamed El Bachraoui

Second Advisor

Dr. Kanat Abdukhalikov

Third Advisor

Dr. James Griffin

Abstract

In this report we shall introduce q-series and we shall discuss some of their applications to the integer partitions, the sums of squares, and the binomial coefficients. We will present the basic theory of q-series including the most famous theorems and rules governing these objects such as the q-binomial theorem and the Jacobi’s triple identity. We shall present the q-binomial coefficients which roughly speaking connect the binomial coefficients to q-series, we will give the most important results on q-binomial coefficients, and we shall provide some of our new results on the divisibility of binomial coefficients. Moreover, we shall give some well-known applications of q-series to sums of two squares and to integer partitions such as Ramanujan’s modulo 5 congruence.

Included in

Mathematics Commons

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