no code implementations • 2 Dec 2017 • Mircea Merca, Maxie D. Schmidt
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 $L_a(\alpha, \beta, q) := \sum_{n \geq 1} a_n q^{\alpha n-\beta} / (1-q^{\alpha n-\beta})$ for integers $\alpha, \beta$ defined such that $\alpha \geq 1$ and $0 \leq \beta < \alpha$.
Number Theory 11A25, 11P81, 05A17, 05A19
no code implementations • 7 Jun 2017 • Mircea Merca, Maxie D. Schmidt
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).
Combinatorics 11A25, 11P81, 05A17, 05A19
no code implementations • 1 Jun 2017 • Mircea Merca, Maxie D. Schmidt
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.
Number Theory 11A25, 11P81, 05A17, 05A19
1 code implementation • 9 May 2017 • Maxie D. Schmidt
We also consider applications of our new results to asymptotic approximations for sums over these divisor functions and to the forms of perfect numbers defined by the special case of the divisor function, $\sigma(n)$, when $\alpha := 1$.
Number Theory
no code implementations • 18 Apr 2017 • Maxie D. Schmidt
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.
Number Theory
no code implementations • 8 Dec 2016 • Maxie D. Schmidt
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 $\sigma_1(n)$, 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.
Number Theory
no code implementations • 3 Nov 2016 • Maxie D. Schmidt
We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function.
Combinatorics Number Theory
no code implementations • 30 Oct 2016 • Maxie D. Schmidt
We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums.
Combinatorics Number Theory
no code implementations • 30 Oct 2016 • Maxie D. Schmidt
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.
Combinatorics
1 code implementation • 9 Sep 2016 • Maxie D. Schmidt
The results proved in the article lead to new applications and integral representations for special function series, sequence generating functions, and other related applications.
Number Theory 05A15, 44A99, 33E99, 11B73
