## RESEARCH INTERESTS

My research interests are primarily in number theory and enumerative combinatorics with an emphasis on generating function methods, continued fractions, software development, and experimental mathematics.
Research problems I have historically been interested in include: combinatorial properties of generalized factorial functions and their symbolic polynomial expansions, generating functions and generating function transformations, generalized Stirling numbers, Jacobi-type continued fractions (J-fractions) for special series, and Lambert series generating functions.

I am always open to exploring new interesting problems. My favorite new exact formula for the generalized sum-of-divisors functions is stated below as follows:

## EDUCATION

### 2014

University of Illinois at Urbana-Champaign

*Master of Science in Computer Science*

### 2012

University of Illinois at Urbana-Champaign

*Bachelor of Science in Mathematics*

### 2012

University of Illinois at Urbana-Champaign

*Bachelor of Science in Computer Science*

## AWARDS AND HONORS

### 2014

NSF GRFP National Honorable Mention

### 2013

NSF GRFP National Honorable Mention

### 2010

Barry M. Goldwater Scholarship

## PUBLICATIONS

### 2018

A Partition Identity Related to Stanley's Theorem

*With Mircea Merca. Accepted for publication in the American Mathematical Monthly, to appear in 2018.*

__Keywords:__ Lambert series; Euler’s totient; partition.
__MSC (2010):__ 11A25; 11P81; 05A17; 05A19.

### 2017

Continued Fractions for Square Series Generating Functions

*Accepted for publication in the Ramanujan Journal (to appear)*

__Full Text:__ https://arxiv.org/abs/1612.02778 (math.NT)
__Keywords:__ square series; q-series; J-fraction; continued fraction; sum of squares functions; sum of divisors function; theta function; ordinary generating function.
__MSC (2010):__ 05A15; 11Y65; 11B65; 40A15.

### 2017

New Recurrence Relations and Matrix Equations for Arithmetic Functions Generated by Lambert Series

*Accepted for publication in Acta Arithmetica (to appear)*

### 2017

Continued Fractions and q-Series Generating Functions for the Generalized Sum-of-Divisors Functions

*Published in the Journal of Number Theory*

### 2017

Generating Function Transformations Related to Polylogarithm Functions and the k-Order Harmonic Numbers

*Published in the Online Journal of Analytic Combinatorics*

### 2017

Square Series Generating Function Transformations

*Published in the Journal of Inequalities and Special Functions*

### 2017

Jacobi-type continued fractions for the ordinary generating functions of generalized factorial functions

*Published in the Journal of Integer Sequences*

### 2014

A Computer Algebra Package for Polynomial Sequence Recognition

*UIUC Master's Thesis*

### 2010

Generalized j-factorial functions, polynomials, and applications

*Published in the Journal of Integer Sequences*

## MANUSCRIPTS

### 08/2017

Factorization Theorems for Hadamard Products and Higher-Order Derivatives of Lambert Series Generating Functions

__Keywords:__ Lambert series; factorization theorem; matrix factorization; partition function; Hadamard product.
__MSC (2010):__ 11A25; 11P81; 05A17; 05A19.

### 07/2017

Factorization Theorems for Generalized Lambert Series and Applications

*Joint work with Mircea Merca.*
__Keywords:__ Lambert series; factorization theorem; matrix factorization; partition function; multiplicative function.
__MSC (2010):__ 11A25; 11P81; 05A17; 05A19.

### 07/2017

Pair Correlation and Gap Distributions for Substitution Tilings and Generalized Ulam Sets in the Plane

*Worked closely with Jayadev Athreya on the project.*
__Full Text:__ https://arxiv.org/abs/1707.05509
__Keywords:__ substitution tiling; Ammann chair; Ulam set; directional distribution;
gap distribution; pair correlation.
__MSC (2010):__ 52C20; 06A99; 11B05; 62H11; 52C23.

### 06/2017

New Factor Pairs for Factorizations of Lambert Series Generating Functions

*With Mircea Merca. *
__Full Text:__ https://arxiv.org/abs/1706.02359
__Keywords:__ Lambert series; factorization theorem; matrix factorization; partition function.
__MSC (2010):__ 11A25; 11P81; 05A17; 05A19.

### 06/2017

Generating Special Arithmetic Functions by Lambert Series Factorizations

*With Mircea Merca. *
__Full Text:__ https://arxiv.org/abs/1706.00393
__Keywords:__ Lambert series; factorization theorem; matrix factorization; partition function.
__MSC (2010):__ 11A25; 11P81; 05A17; 05A19.

### 05/2017

Exact Formulas for the Generalized Sum-of-Divisors Functions

__Full Text:__ https://arxiv.org/abs/1705.03488 (math.NT)
__Keywords:__ divisor function; sum-of-divisors function; Lambert series; perfect number.
__MSC (2010):__ 30B50; 11N64; 11B83.

### 04/2017

Combinatorial Sums and Identities Involving Generalized Divisor Functions with Bounded Divisors

__Full Text:__ https://arxiv.org/abs/1704.05595 (math.NT)
__Keywords:__ divisor function; sum of divisors function; Lambert series.
__MSC (2010):__ 30B50; 11N64; 11B37; 11B83; 11K65.

### 02/2017

Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers h ≥ 2

__Full Text:__ https://arxiv.org/abs/1702.01374 (math.CO)

### 01/2017

New Congruences and Finite Difference Equations for Generalized Factorial Functions

__Full Text:__ https://arxiv.org/abs/1701.04741 (math.CO)
__Keywords:__ continued fraction, Pochhammer symbol, multiple factorial, Pochhammer
k-symbol, prime congruence, Wilson’s theorem, Clement’s theorem, sexy prime.
__MSC (2010):__ 05A10; 11Y55; 11Y65; 11A07; 11B50.

### 11/2016

Combinatorial Identities for Generalized Stirling Numbers Expanding f-Factorial Functions and the f-Harmonic Numbers

*Summary of my CS499 Senior Thesis project from 2010-2011 at UIUC.*
__Full Text:__ https://arxiv.org/abs/1611.04708 (math.CO)
__Keywords:__ factorial; multifactorial; j-factorial; Pochhammer symbol; Stirling number; generalized Stirling number; harmonic number; f-harmonic number; Stirling polynomial.
__MSC (2010):__ 11B73; 05A10; 11B75.

### 11/2016

Zeta Series Generating Function Transformations Related to Generalized Stirling Numbers and Partial Sums of the Hurwitz Zeta Function

__Full Text:__ https://arxiv.org/abs/1611.00957 (math.CO)