Title: | On sequences related to binary partition function and the Thue-Morse sequence |
Author(s): | Ekvittayaniphon, Sakulbuth |
Director of Research: | Reznick, Bruce |
Doctoral Committee Chair(s): | Hildebrand, A.J. |
Doctoral Committee Member(s): | Berndt, Bruce C.; Boca, Florin P. |
Department / Program: | Mathematics |
Discipline: | Mathematics |
Degree Granting Institution: | University of Illinois at Urbana-Champaign |
Degree: | Ph.D. |
Genre: | Dissertation |
Subject(s): | Binary Partition, Function, Thue-Morse Sequence |
Abstract: | In this dissertation, we discuss properties of the family of sequences $\mathbf{u_d} = \{u_d(n)\}_{n \geq 0}$ for positive integer $d$. We define them by letting $u_d(n)$ be the coefficient of $X^n$ in $\displaystyle \prod_{j=0}^{\infty} \left( 1 - X^{2^j}\right) \left( 1 - X^{d\cdot 2^j}\right)^{-1}$. First, we discuss the binary partition function and its relationship with the sequence $\mathbf{u_d}$. We then give several intermediate results and identities. Afterward, we generalize the sequence with different initial values. We also look at the corresponding generating function. After this, we focus on its asymptotic behavior by illustrating the cases when $d=3,5,9$. Finally, we explain asymptotic behavior for general cases and establish conjectures based on numerical data.
Then, we investigate another family of sequences, $\mathbf{x_k} = \{x_k(n)\}_{n \geq 0}$, defined by $x_k(n) = |t_{n+k} - t_n|$ where $\textbf{t} = \{t_n\}_{n \geq 0}$ is the Thue-Morse sequence. We give the frequency of $1$'s and $0$'s of each sequence $\mathbf{x_k}$ and express them in terms of recurrence relations. We note the similarity with the Stern sequence, denoted by $\textbf{s} = \{s(n)\}_{n \geq 0}$ . Further, we investigate the frequency of appearances of $00$, $01$, $10$, and $11$ of each sequence.
Finally, we define the correlation function related to the sequence $\mathbf{x_k}$, denoted by $f(d)$, and the associated density function $\tilde{f}(d)$. We present both recurrence relations, and closed formulas for values of $d$ near powers of 2. |
Issue Date: | 2018-07-13 |
Type: | Text |
URI: | http://hdl.handle.net/2142/101563 |
Rights Information: | Copyright 2018 Sakulbuth Ekvittayaniphon |
Date Available in IDEALS: | 2018-09-27 |
Date Deposited: | 2018-08 |