ATL Family Portrait: images/maxie-brandye-2015.png

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)

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