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.
Recommended Citation
Elewoday, Mohamed Ammar, "Algebraic, Analytic, and Combinatorial Properties of Power Product Expansions in Two Independent Variables." (2021). Graduate Theses, Dissertations, and Problem Reports. 8308.
https://researchrepository.wvu.edu/etd/8308
Included in
Analysis Commons, Discrete Mathematics and Combinatorics Commons, Harmonic Analysis and Representation Commons
