Semester

Summer

Date of Graduation

2021

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Harry Gingold

Committee Co-Chair

Jocelyn Quaintance

Committee Member

Harumi Hattori

Committee Member

Rong Luo

Committee Member

Jerzy Wojciechowski

Abstract

Let $F(x,y)=I+\hspace{-.3cm}\sum\limits_{\substack{p=1\\m+n=p}}^{\infty}\hspace{-.3cm}A_{m,n}x^my^n$ be a formal power series, where the coefficients $A_{m,n}$ are either all matrices or all scalars. We expand $F(x,y)$ into the formal products $\prod\limits_{\substack{p=1\\m+n=p}}^{\infty}\hspace{-.3cm}(I+G_{m,n}x^m y^n)$, $\prod\limits_{\substack{p=1\\m+n=p}}^{\infty}\hspace{-.3cm}(I-H_{m,n}x^m y^n)^{-1}$, namely the \textit{ power product expansion in two independent variables} and \textit{inverse power product expansion in two independent variables} respectively. By developing new machinery involving the majorizing infinite product, we provide estimates on the domain of absolute convergence of the infinite product via the Taylor series coefficients of $F(x,y)$. This machinery introduces a myriad of "mixed expansions", uncovers various algebraic connections between the $(A_{m,n})$ and the $(G_{m,n})$, and uncovers various algebraic connections between the $(A_{m,n})$ and the $(H_{m,n})$, and leads to the identification of the domain of absolute convergence of the power product and the inverse power product as a Cartesian product of polydiscs. This makes it possible to use the truncated power product expansions $\prod\limits_{\substack{p=1\\m+n=p}}^{P}(1+G_{m,n}x^my^n)$,$\prod\limits_{\substack{p=1\\m+n=p}}^{P}(1-H_{m,n}x^my^n)^{-1}$ as approximations to the analytic function $F(x,y)$. The results are made possible by certain algebraic properties characteristic of the expansions. Moreover, in the case where the coefficients $A_{m,n}$ are scalars, we derive two asymptotic formulas for the $G_{m,n},H_{m,n}$, with $m$ {\it fixed}, associated with the majorizing power series. We also discuss various combinatorial interpretations provided by these power product expansions.

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