Related papers: The Catalan Equation in Finitely Generated Domains

In this note we show that various natural q-analogues of the Catalan numbers can be obtained in a uniform way. Furthermore we compute their Hankel determinants.

We study a refinement of the $q,t$-Catalan numbers introduced by Xin and Zhang (2022, 2023) using tools from polyhedral geometry. These refined $q,t$-Catalan numbers depend on a vector of parameters $\vec{k}$ and the classical $q,t$-Catalan…

Let $f:\mathbb{Z}\longrightarrow \{ \times \cdot\}$ be a function such that $f(a) = \cdot$ for all except finitely for many $a \in \mathbb{Z}$. We define a set $\flat f$ of non-intersecting arc (or cap) diagrams satisfying certain…

We prove some existence and nonexistence results for a class of critical $(p,q)$-Laplacian problems in a bounded domain. Our results extend and complement those in the literature for model cases.

We consider nonhomogeneous fractional $p$-Laplace equations defined on a bounded nonsmooth domain which goes beyond the Lipschitz category. Under a sufficient flatness assumption on the domain in the sense of Reifenberg, we establish…

For each integer $k\ge 1$, we define an algorithm which associates to a partition whose maximal value is at most $k$ a certain subset of all partitions. In the case when we begin with a partition $\lambda$ which is square, i.e…

Let $p$ be a real number greater than one and let $G$ be a finitely generated, infinite group. In this paper we introduce the $p$-harmonic boundary of $G$. We then characterize the vanishing of the first reduced $\ell^p$-cohomology of $G$…

We present existence and nonexistence results on the solution of an overdetermined problem for the normalized p-Laplacian in a bounded open set, with p ranging from 1 to infinity. More precisely we consider a non-constant Neumann condition…

In this paper, we investigate the existence of infinitely many solutions for the following elliptic boundary value problem with $(p,q)$-Kirchhoff type \begin{eqnarray*} \begin{cases} -\Big[M_1\left(\int_\Omega|\nabla u_1|^p…

We prove a central limit theorem for the joint distribution of $s_q(A_jn)$, $1\le j \le d$, where $s_q$ denotes the sum-of-digits function in base~$q$ and the $A_j$'s are positive integers relatively prime to $q$. We do this in fact within…

Denote the set of algebraic numbers as $\overline{\mathbb{Q}}$ and the set of algebraic integers as $\overline{\mathbb{Z}}$. For $\gamma\in\overline{\mathbb{Q}}$, consider its irreducible polynomial in $\mathbb{Z}[x]$,…

In a recent letter, new representations were proposed for the pair of sequences ($\gamma,\delta$), as defined formally by Bailey in his famous lemma. Here we extend and prove this result, providing pairs ($\gamma,\delta$) labelled by the…

Let p/q be a rational number. Numeration in base p/q is defined by a function that evaluates each finite word over A_p={0,1,...,p-1} to some rational number. We let N_p/q denote the image of this evaluation function. In particular, N_p/q…

In this note, we give an alternative proof of the following result. Let p, q >= 2 be two multiplicatively independent integers. If an infinite set of integers is both p- and q-recognizable, then it is syndetic. Notice that this result is…

In the article we study the Neumann $(p,q)$-eigenvalue problems in bounded H\"older $\gamma$-singular domains $\Omega_{\gamma}\subset \mathbb{R}^n$. In the case $1<p<\infty$ and $1<q<p^{*}_{\gamma}$ we prove solvability of this eigenvalue…

Recently, Agievich proposed an interesting upper bound on binomial coefficients in the de Moivre-Laplace form. In this article, we show that the latter bound, in the specific case of a central binomial coefficient, is larger than the one…

Given two polynomials $P,q$ we consider the following question: "how large can the index of the first non-zero moment $\tilde{m}_k=\int_a^b P^k q$ be, assuming the sequence is not identically zero?". The answer $K$ to this question is known…

The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns \cite{Raney}. We give a new combinatorial interpretation for all…

The purpose of this paper is to study H\"older estimates for the $\bar\partial$ problem for $(p,q)$ forms on products of general planar domains. As indicated by an example of Stein and Kerzman, solutions to the $\bar\partial$ problem on…

In this article we will prove that if the continuous closed curve $\gamma : [0, 1] \rightarrow \mathbb{R}^2$ has finite $p$-variation with $p < 2$, then $(\iint\limits_{\mathbb{R}^2}|\eta(\gamma, (x, y))|^q \,dx \,dy)^{1/q} \le…