Some qq-series identities extending work of Andrews, Crippa, and Simon on sums of divisors functions

In this article we extend a theorem of Andrews, Crippa, and Simon on the asymptotic behavior of polynomials defined by a general class of recursive equations. Here the polynomials are in the variable $q$, and the recursive definition at step $n$ introduces a polynomial in $n$. Our extension replaces the polynomial in $n$ with either an exponential or periodic function of $n$

Extensions of MacMahon's sums of divisors

In 1920, P. A. MacMahon generalized the (classical) notion of divisor sums by relating it to the theory of partitions of integers. In this paper, we extend the idea of MacMahon. In doing so we reveal a wealth of divisibility theorems and unexpected combinatorial identities. Our initial approach is quite different from MacMahon and involves rational function approximation to MacMahon-type generating functions. One such example involves multiple $q$-harmonic sums $$\sum_{k=1}^n\frac{(-1)^{k-1}\genfrac{[}{]}{0pt}{}{n}{k}_{q}(1+q^k)q^{\binom{k}{2}+tk}}{[k]_q^{2t} \genfrac{[}{]}{0pt}{}{n+k}{k}_{q}}=\sum_{1\leq k_1\leq\cdots\leq k_{2t}\leq n}\frac{q^{n+k_1+k_3\cdots+k_{2t-1}}+q^{k_2+k_4+\cdots+k_{2t}}}{[n+k_1]_q[k_2]_q\cdots[k_{2t}]_q}.$

Polynomials: Special Polynomials and Number-Theoretical Applications

Polynomials play a crucial role in many areas of mathematics including algebra, analysis, number theory, and probability theory. They also appear in physics, chemistry, and economics. Especially extensively studied are certain infinite families of polynomials. Here, we only mention some examples: Bernoulli, Euler, Gegenbauer, trigonometric, and orthogonal polynomials and their generalizations. There are several approaches to these classical mathematical objects. This Special Issue presents nine high quality research papers by leading researchers in this field. I hope the reading of this work will be useful for the new generation of mathematicians and for experienced researchers as wel

q-Series Arising from the Study of Random Graphs

. This paper deals with q--series arising from the study of the transitive closure problem in random acyclic digraphs. In particular it presents an identity involving divisor generating functions which allows to determine the asymptotic behavior of polynomials defined by a general class of recursive equations, including the polynomials for the mean and the variance of the size of the transitive closure in random acyclic digraphs. Key words. q--series, divisor generating functions, polynomials, random graphs, transitive closure, probability distributions. AMS subject classifications. 05C80, 11P81, 33D15, 60E10, 60F99 1. Introduction. In [7] Simon, Crippa and Collenberg studied the distribution of the transitive closure in the G n;p --model of a random acyclic digraph. By interpreting the random variable describing the size of the transitive closure of a node as a discrete--time, pure--birth process, they succeeded in finding closed expressions for its distribution, mean and variance...