arXiv subjects

Maxie D. Schmidt

Publications and source records attributed to Maxie D. Schmidt.

At least 19 recordsLinked to original sources

arXiv2023

The partition function $p(n)$ in terms of the classical Möbius function

In this paper, we investigate decompositions of the partition function $p(n)$ from the additive theory of partitions considering the famous Möbius function $μ(n)$ from multiplicative number theory. Some combinatorial interpretations are given in this context. Our work extends several analogous identities proved recently relating $p(n)$ and Euler's totient function $φ(n)$. Keywords: Lambert series; Möbius function; $q$-series; partition function

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arXiv2023

A Partition Identity Related to Stanley's Theorem

In this paper, we use the Lambert series generating function for Euler's totient function to introduce a new identity for the number of $1$'s in the partitions of $n$. A new expansion for Euler's partition function $p(n)$ is derived in this context. These surprising new results connect the famous classical totient function from multiplicative number theory to the additive theory of partitions.

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arXiv2020

Combinatorial Sums and Identities Involving Generalized Sum-of-Divisors Functions with Bounded Divisors

The class of Lambert series generating functions (LGFs) denoted by $L_α(q)$ formally enumerate the generalized sum-of-divisors functions, $σ_α(n) = \sum_{d|n} d^α$, for all integers $n \geq 1$ and fixed real-valued parameters $α\geq 0$. We prove new formulas expanding the higher-order derivatives of these LGFs. The results we obtain are combined to express new identities expanding the generalized sum-of-divisors functions. These new identities are expanded in the form of sums of polynomially scaled multiples of a related class of divisor sums depending on $n$ and $α$.

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arXiv2019

Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums

We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithmetic functions $f$ and $g$. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function $p(n)$ to arbitrary arithmetic functions $f$. We also conclude the last section of the article by directly expanding new formulas for an arithmetic function $g$ by the type II sums using discrete Fourier transforms for functions over inputs of greatest common divisors and by suitably defined orthogonal polynomial sequences whose weight function we can define by a discrete time Fourier transform (DTFT) involving the partition function $p(n)$. There are numerous applications and special cases of our new results which we are able to cite as examples in the article. Particular cases of the applications we give in the article include new identities for Euler's totient function, the Ramanujan sums $c_q(n)$, the generalized sum-of-divisors functions, the Mertens function which is the summatory function of the Möbius function, and the cyclotomic polynomials.

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arXiv2019

A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions

The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence's ordinary and exponential generating function (OGF and EGF, respectively) and vice versa. The Laplace transform provides an integral formula for the EGF-to-OGF transformation, where the reverse OGF-to-EGF operation requires more careful integration techniques. We prove two variants of the OGF-to-EGF transformation integrals from the Hankel loop contour for the reciprocal gamma function and from Fourier series expansions of integral representations for the Hadamard product of two generating functions, respectively. We also suggest several generalizations of these integral formulas and provide new examples along the way.

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arXiv2019

Exact Formulas for the Generalized Sum-of-Divisors Functions

We prove new exact formulas for the generalized sum-of-divisors functions, $σ_α(x) := \sum_{d|x} d^α$. The formulas for $σ_α(x)$ when $α\in \mathbb{C}$ is fixed and $x \geq 1$ involves a finite sum over all of the prime factors $n \leq x$ and terms involving the $r$-order harmonic number sequences and the Ramanujan sums $c_d(x)$. The generalized harmonic number sequences correspond to the partial sums of the Riemann zeta function when $r > 1$ and are related to the generalized Bernoulli numbers when $r \leq 0$ is integer-valued. A key part of our new expansions of the Lambert series generating functions for the generalized divisor functions is formed by taking logarithmic derivatives of the cyclotomic polynomials, $Φ_n(q)$, which completely factorize the Lambert series terms $(1-q^n)^{-1}$ into irreducible polynomials in $q$. We focus on the computational aspects of these exact expressions, including their interplay with experimental mathematics, and comparisons of the new formulas for $σ_α(n)$ and the summatory functions $\sum_{n \leq x} σ_α(n)$. Keywords: divisor function; sum-of-divisors function; Lambert series; perfect number. MSC (2010): 30B50; 11N64; 11B83

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arXiv2017

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

We first summarize joint work on several preliminary canonical Lambert series factorization theorems. Within this article we establish new analogs to these original factorization theorems which characterize two specific primary cases of the expansions of Lambert series generating functions: factorizations for Hadamard products of Lambert series and for higher-order derivatives of Lambert series. The series coefficients corresponding to these two generating function cases are important enough to require the special due attention we give to their expansions within the article, and moreover, are significant in that they connect the characteristic expansions of Lambert series over special multiplicative functions to the explicitly additive nature of the theory of partitions. Applications of our new results provide new exotic sums involving multiplicative functions, new summation-based interpretations of the coefficients of the integer-order $j^{th}$ derivatives of Lambert series generating functions, several new series for the Riemann zeta function, and an exact identity for the number of distinct primes dividing $n$.

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arXiv2017

Factorization Theorems for Generalized Lambert Series and Applications

We prove new variants of the Lambert series factorization theorems studied by Merca and Schmidt (2017) which correspond to a more general class of Lambert series expansions of the form $L_a(α, β, q) := \sum_{n \geq 1} a_n q^{αn-β} / (1-q^{αn-β})$ for integers $α, β$ defined such that $α\geq 1$ and $0 \leq β< α$. Applications of the new results in the article are given to restricted divisor sums over several classical special arithmetic functions which define the cases of well-known, so-termed "ordinary" Lambert series expansions cited in the introduction. We prove several new forms of factorization theorems for Lambert series over a convolution of two arithmetic functions which similarly lead to new applications relating convolutions of special multiplicative functions to partition functions and $n$-fold convolutions of one of the special functions.

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arXiv2017

Generating Special Arithmetic Functions by Lambert Series Factorizations

We summarize the known useful and interesting results and formulas we have discovered so far in this collaborative article summarizing results from two related articles by Merca and Schmidt arriving at related so-termed Lambert series factorization theorems. We unify the matrix representations that underlie two of our separate papers, and which commonly arise in identities involving partition functions and other functions generated by Lambert series. We provide a number of properties and conjectures related to the inverse matrix entries defined in Schmidt's article and the Euler partition function $p(n)$ which we prove through our new results unifying the expansions of the Lambert series factorization theorems within this article.

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arXiv2017

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

We study empirical statistical and gap distributions of several important tilings of the plane. In particular, we consider the slope distributions, the angle distributions, pair correlation, squared-distance pair correlation, angle gap distributions, and slope gap distributions for the Ammann Chair tiling, the recently discovered fifteenth pentagonal tiling, and a few pertinent tilings related to these famous examples. We also consider the spatial statistics of generalized Ulam sets in two dimensions. Additionally, we carefully prove a tight asymptotic formula for the time steps in which Ulam set points at certain prescribed geometric positions in their plots in the plane formally enter the recursively-defined sets. The software we have developed to these generate numerical approximations to the distributions for the tilings we consider here is written in Python under the Sage environment and is released as open-source software which is available freely on our websites. In addition to the small subset of tilings and other point sets in the plane we study within the article, our program supports many other tiling variants and is easily extended for researchers to explore related tilings and iterative sets.

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arXiv2017

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

We consider relations between the pairs of sequences, $(f, g_f)$, generated by the Lambert series expansions, $L_f(q) = \sum_{n \geq 1} f(n) q^n / (1-q^n)$, in $q$. In particular, we prove new forms of recurrence relations and matrix equations defining these sequences for all $n \in \mathbb{Z}^{+}$. The key ingredient to the proof of these results is given by the statement of Euler's pentagonal number theorem expanding the series for the infinite $q$-Pochhammer product, $(q; q)_{\infty}$, and for the first $n$ terms of the partial products, $(q; q)_n$, forming the denominators of the rational $n^{th}$ partial sums of $L_f(q)$. Examples of the new results given in the article include new exact formulas for and applications to the Euler phi function, $ϕ(n)$, the Möbius function, $μ(n)$, the sum of divisors functions, $σ_1(n)$ and $σ_α(n)$, for $α\geq 0$, and to Liouville's lambda function, $λ(n)$.

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arXiv2017

New Factor Pairs for Factorizations of Lambert Series Generating Functions

We prove several new variants of the Lambert series factorization theorem established in the first article "Generating special arithmetic functions by Lambert series factorizations" by Merca and Schmidt (2017). Several characteristic examples of our new results are presented in the article to motivate the formulations of the generalized factorization theorems. Applications of these new factorization results include new identities involving the Euler partition function and the generalized sum-of-divisors functions, the Möbius function, Euler's totient function, the Liouville lambda function, von Mangoldt's lambda function, and the Jordan totient function.

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arXiv2017

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

We construct new continued fraction expansions of Jacobi-type J-fractions in $z$ whose power series expansions generate the ratio of the $q$-Pochhamer symbols, $(a; q)_n / (b; q)_n$, for all integers $n \geq 0$ and where $a,b,q \in \mathbb{C}$ are non-zero and defined such that $|q| < 1$ and $|b/a| < |z| < 1$. If we set the parameters $(a, b) := (q, q^2)$ in these generalized series expansions, then we have a corresponding J-fraction enumerating the sequence of terms $(1-q) / (1-q^{n+1})$ over all integers $n \geq 0$. Thus we are able to define new $q$-series expansions which correspond to the Lambert series generating the divisor function, $d(n)$, when we set $z \mapsto q$ in our new J-fraction expansions. By repeated differentiation with respect to $z$, we also use these generating functions to formulate new $q$-series expansions of the generating functions for the sums-of-divisors functions, $σ_α(n)$, when $α\in \mathbb{Z}^{+}$. To expand the new $q$-series generating functions for these special arithmetic functions we define a generalized classes of so-termed Stirling-number-like "$q$-coefficients", or Stirling $q$-coefficients, whose properties, relations to elementary symmetric polynomials, and relations to the convergents to our infinite J-fractions are also explored within the results proved in the article.

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arXiv2017

Square Series Generating Function Transformations

We construct new integral representations for transformations of the ordinary generating function for a sequence, $\langle f_n \rangle$, into the form of a generating function that enumerates the corresponding "square series" generating function for the sequence, $\langle q^{n^2} f_n \rangle$, at an initially fixed non-zero $q \in \mathbb{C}$. The new results proved in the article are given by integral-based transformations of ordinary generating function series expanded in terms of the Stirling numbers of the second kind. We then employ known integral representations for the gamma and double factorial functions in the construction of these square series transformation integrals. The results proved in the article lead to new applications and integral representations for special function series, sequence generating functions, and other related applications. A summary Mathematica notebook providing derivations of key results and applications to specific series is provided online as a supplemental reference to readers.

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arXiv2017

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

We introduce a class of $f(t)$-factorials, or $f(t)$-Pochhammer symbols, that includes many, if not most, well-known factorial and multiple factorial function variants as special cases. We consider the combinatorial properties of the corresponding generalized classes of Stirling numbers of the first kind which arise as the coefficients of the symbolic polynomial expansions of these $f$-factorial functions. The combinatorial properties of these more general parameterized Stirling number triangles we prove within the article include analogs to known expansions of the ordinary Stirling numbers by $p$-order harmonic number sequences through the definition of a corresponding class of $p$-order $f$-harmonic numbers.

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arXiv2017

Continued Fractions for Square Series Generating Functions

We consider new series expansions for variants of the so-termed ordinary geometric square series generating functions originally defined in the recent article titled "Square Series Generating Function Transformations" (arXiv: 1609.02803). Whereas the original square series transformations article adapts known generating function transformations to construct integral representations for these square series functions enumerating the square powers of $q^{n^2}$ for some fixed non-zero $q$ with $|q| < 1$, we study the expansions of these special series through power series generated by Jacobi-type continued fractions, or J-fractions. We prove new exact expansions of the $h^{th}$ convergents to these continued fraction series and show that the limiting case of these convergent generating functions exists. We also prove new infinite $q$-series representations of special square series expansions involving square-power terms of the series parameter $q$, the $q$-Pochhammer symbol, and double sums over the $q$-binomial coefficients. Applications of the new results we prove within the article include new $q$-series representations for the ordinary generating functions of the special sequences, $r_p(n)$, and $σ_1(n)$, as well as parallels to the examples of the new integral representations for theta functions, series expansions of infinite products and partition function generating functions, and related unilateral special function series cited in the first square series transformations article.

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arXiv2017

Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers $h \geq 2$

We prove two new forms of Jacobi-type J-fraction expansions generating the binomial coefficients, $\binom{x+n}{n}$ and $\binom{x}{n}$, over all $n \geq 0$. Within the article we establish new forms of integer congruences for these binomial coefficient variations modulo any (prime or composite) $h \geq 2$ and compare our results with existing known congruences for the binomial coefficients modulo primes $p$ and prime powers $p^k$. We also prove new exact formulas for these binomial coefficient cases from the expansions of the $h^{th}$ convergent functions to the infinite J-fraction series generating these coefficients for all $n$.

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arXiv2017

Jacobi Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions

The article studies a class of generalized factorial functions and symbolic product sequences through Jacobi type continued fractions (J-fractions) that formally enumerate the divergent ordinary generating functions of these sequences. The more general definitions of these J-fractions extend the known expansions of the continued fractions originally proved by Flajolet that generate the rising factorial function, or Pochhammer symbol, $(x)_n$, at any fixed non-zero indeterminate $x \in \mathbb{C}$. The rational convergents of these generalized J-fractions provide formal power series approximations to the ordinary generating functions that enumerate many specific classes of factorial-related integer product sequences. The article also provides applications to a number of specific identities, new integer congruence relations satisfied by generalized factorial-related product sequences and the $r$-order harmonic numbers, among several other notable motivating examples as immediate applications of the new results. In this sense, the article serves as a semi-comprehensive, detailed survey reference that introduces applications to many established and otherwise well-known combinatorial identities, new cases of generating functions for factorial-function-related product sequences, and other examples of the generalized integer-valued multifactorial, or $α$-factorial, function sequences. The convergent-based generating function techniques illustrated by the particular examples cited within the article are easily extended to enumerate the factorial-like product sequences arising in the context of many other specific applications.

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