Related papers: Partition-theoretic Frobenius-type limit formulas

The coefficients of the generating function $(q;q)^\alpha_\infty$ produce $p_\alpha(n)$ for $\alpha \in \mathbb{Q}$. In particular, when $\alpha = -1$, the partition function is obtained. Recently, Chan and Wang identified and proved…

We derive a functional central limit theorem (fclt) for normalised sums of a function of the partial sums of independent and identically distributed random variables. In particular, we show, using a technique presented in Huang and Zhang…

Let $p_{k}(n)$ be the coefficient of $q^n$ in the series expansion of $(q;q)_{\infty}^{k}$. It is known that the partition function $p(n)$, which corresponds to the case when $k=-1$, satisfies congruences such as $p(5n+4)\equiv 0\pmod{5}$.…

If the odd and even parts of a continued fraction converge to different values, the continued fraction may or may not converge in the general sense. We prove a theorem which settles the question of general convergence for a wide class of…

Let $A$ be an abelian variety over a finite field $k$ with $|k|=q=p^m$. Let $\pi\in \text{End}_k(A)$ denote the Frobenius and let $v=\frac{q}{\pi}$ denote Verschiebung. Suppose the Weil $q$-polynomial of $A$ is irreducible. When…

In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, $\phi_k(n)$ and $c\phi_k(n),$ which enumerate two types of combinatorial objects which Andrews called generalized Frobenius…

Let $ \mathbb{Q}\mathcal{E}_{\mathbb{Z}} $ be the set of power sums whose characteristic roots belong to $ \mathbb{Z} $ and whose coefficients belong to $ \mathbb{Q} $, i.e. $ G : \mathbb{N} \rightarrow \mathbb{Q} $ satisfies…

Let $f:\Z/q\Z\rightarrow\Z$ be such that $f(a)=\pm 1$ for $1\le a<q$, and $f(q)=0$. Then Erd\"os conjectured that $\sum_{n\ge1}\frac{f(n)}{n} \ne 0$. For $q$ even, this is trivially true. If $q\equiv 3$ ( mod $4$), Murty and Saradha proved…

We study for bounded multiplicative functions $f$ sums of the form \begin{align*} \sum_{\substack{n\leq x \atop n\equiv a\pmod q}}f(n), \end{align*} establishing that their variance over residue classes $a \pmod q$ is small as soon as…

Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a…

By making use of the multiplicate form of the extended Carlitz inverse series relations, we establish two general `dual' theorems of Jackson's summation formula for well--poised $_8\phi_7$-series. Their duplicate forms under the partition…

Frobenius built a representation theory of finite groups in the process of obtaining the irreducible factorization of the group determinant. Here, we give a generalization of Frobenius' theorem. The generalization leads to a corollary on…

The quantum integer [n]_q is the polynomial 1 + q + q^2 + ... + q^{n-1}, and the sequence of polynomials { [n]_q }_{n=1}^{\infty} is a solution of the functional equation f_{mn}(q) = f_m(q)f_n(q^m). In this paper, semidirect products of…

We prove that if $q_1, \ldots, q_m: {\Bbb R}^n \longrightarrow {\Bbb R}$ are quadratic forms in variables $x_1, \ldots, x_n$ such that each $q_k$ depends on at most $r$ variables and each $q_k$ has common variables with at most $r$ other…

We use an elementary argument to prove some finite sums involving expressions of the forms $(q)_n$ and $(a;q)_n$ along with inductive formulas for some sequences.

We generalize and prove a hypothesis by V. Arnold on the parity of Frobenius number. For the case of symmetric semigroups with three generators of Frobenius numbers we found an exact formula, which in a sense is the sum of two Sylvester's…

We provide numerical indications of the $q$-generalised central limit theorem that has been conjectured (Tsallis 2004) in nonextensive statistical mechanics. We focus on $N$ binary random variables correlated in a {\it scale-invariant} way.…

Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1).…

For real power series whose non-zero coefficients satisfy $|a_m|^{1/m}\to~1$ we prove a stronger version of Fabry theorem relating the frequency of sign changes in the coefficients and analytic continuation of the sum of the power series.

A generalization of a beautiful $q$-series identity found in the unorganized portion of Ramanujan's second and third notebooks is obtained. As a consequence, we derive a new three-parameter identity which is a rich source of…