相关论文: q-Catalan numbers and q-Narayana polynomials

For basic discrete probability distributions, $-$ Bernoulli, Pascal, Poisson, hypergeometric, contagious, and uniform, $-$ $q$-analogs are proposed.

We prove a Lucas-type congruence for q-Delannoy numbers.

We compute the Poincar\'e polynomials of the compactified Jacobians for plane curve singularities with Puiseaux exponents $(nd,md,md+1)$, and relate them to the combinatorics of $q,t$-Catalan numbers in the non-coprime case. We also confirm…

We define the notion of a Catalan pair (which is a pair of binary relations (S,R) satisfying certain axioms) with the aim of giving a common language to most of the combinatorial interpretations of Catalan numbers. We show, in particular,…

We propose an algebraic model of the conjectural triply graded homology of Gukov, Dunfield and Rasmussen for some torus knots. It turns out to be related to the q,t-Catalan numbers of Garsia and Haiman.

It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a…

The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the…

We study three different $q$-analogues of the harmonic numbers. As applications, we present some generating functions involving number theoretical functions and give the $q$-generalization of Gosper's exponential generating function of…

We introduce four q-analogs of the double Laplace transform and prove some of their main properties. Next we show how they can be used to solve some q-functional equations and partial q-differential equations.

In this paper we continue to investigate a certain class of Hankel-like positive definite kernels using their associated orthogonal polynomials. The main result of this paper is about the structure of this kind of kernels.

We define a weighted analog for the multidimensional Catalan numbers, obtain matrix-based recurrences for some of them, and give conditions under which they are periodic. Building on this framework, we introduce two new sequences of…

We answer the question in the title in the negative by providing four proofs.

This article lists all the solution of the Catalan equation $x^p-y^q=1$ for $x,y \in \mathbb{Z}[i]$, when one of the primes $p and q$ is even.

The purpose of this paper is to construct q-Euler numbers and polynomials by using p-adic q-integral equations on Zp. Finally, we will give some interesting formulae related to these q-Euler numbers and polynomials.

We give a simple recursion labeled by binary sequences which computes rational $q,t$-Catalan power series, both in relatively prime and non relatively prime cases. It is inspired by, but not identical to recursions due to B. Elias, M.…

We establish a connection between constructible representations (arising in the study of left cells in Weyl groups) and Catalan numbers.

In this paper we consider the generalized Catalan numbers F(s,n)= 1/((s-1)n+1) binom{sn}{n}. We find all $n$ such that for $p$ prime, p^q divides F(p^q,n), q>=1. As a byproduct we settle a question of Hough and the late Simion on the…

The Hankel determinants $\left(\frac{r}{2(i+j)+r}\binom{2(i+j)+r}{i+j}\right)_{0\leq i,j \leq n-1}$ of the convolution powers of Catalan numbers were considered by Cigler and by Cigler and Krattenthaler. We evaluate these determinants for…

We use ideas from our previous work to obtain some theorems that will allow us to obtain the integer solution of a quadratic polynomial in two variables that represents a natural number

We study the properties of the odd Catalan numbers, C_n, modulo 2^k for k >= 2. We show that there exist only k - 1 different congruences of the odd Catalan numbers modulo 2^k. Moreover, these congruences can be obtained by C_{2^m - 1} (mod…