Related papers: On factorization of q-difference equation for cont…
In this paper, the discriminant of homogeneous polynomials is studied in two particular cases: a single homogeneous polynomial and a collection of n-1 homogeneous polynomials in n variables. In these two cases, the discriminant is defined…
A matrix factorization problem is considered. The matrix to be factorized is algebraic, has dimension 2 X 2 and belongs to Moiseev's class. A new method of factorization is proposed. First, the matrix factorization problem is reduced to a…
Suppose $Q(x)$ is a real $n\times n$ regular symmetric positive semidefinite matrix polynomial. Then it can be factored as $$Q(x) = G(x)^TG(x),$$ where $G(x)$ is a real $n\times n$ matrix polynomial with degree half that of $Q(x)$ if and…
We derive combinatorial formulae for the modified Macdonald polynomial $H_{\lambda}(x;q,t)$ using coloured paths on a square lattice with quasi-cylindrical boundary conditions. The derivation is based on an integrable model associated to…
We show that the quantum invariants arising from typical representations of the quantum group $U_h\mathfrak{sl}(2|1)$ are q-holonomic. In particular, this implies the existence of an underlying field theory for which this family of…
Motivated by the work of Prajapati \emph{et al.} \cite{PAA}, here we study some explicit form of the generalized Laguerre polynomials $L_{\lfloor\frac{n}{q}\rfloor}^{(\alpha,\beta)}(z)$, when $q=1$.
In this paper, we introduce the Rogers-Szeg\"o deformed $q$-differential operators g$_{n}(bD_{q}|u)$ based on $q$-differential operator $D_{q}$. The motivation for introducing the operators g$_{n}(bD_{q})$ is that their limit turns out to…
The plane partition polynomial $Q_n(x)$ is the polynomial of degree $n$ whose coefficients count the number of plane partitions of $n$ indexed by their trace. Extending classical work of E.M. Wright, we develop the asymptotics of these…
Let $\Bbb F_q$ be a finite field with $q$ elements and $n$ a positive integer. Mart\'inez, Vergara and Oliveira \cite{MVO} explicitly factorized $x^{n} - 1$ over $\Bbb F_q$ under the condition of $rad(n)|(q-1)$. In this paper, suppose that…
Given a square, nonsingular matrix of univariate polynomials $\mathbf{F} \in \mathbb{K}[x]^{n \times n}$ over a field $\mathbb{K}$, we give a fast, deterministic algorithm for finding the Hermite normal form of $\mathbf{F}$ with complexity…
For the quantum integer [n]_q = 1+q+q^2+... + q^{n-1} there is a natural polynomial multiplication such that [mn]_q = [m]_q \otimes_q [n]_q. This multiplication is given by the functional equation f_{mn}(q) = f_m(q) f_n(q^m), defined on a…
A hyperbinary partition of the nonnegative integer n is a partition where every part is a power of 2 and every part appears at most twice. We give three applications of the length generating function for such partitions, denoted by h_q(n).…
We consider deformations of the differential of a $q$-differential graded algebra. We prove that it is controlled by a generalized Maurer-Cartan equation. We find explicit formulae for the coefficients $c_k$ involved in that equation.
In this paper, we consider the Carlitz's type q-analogue of Changhee numbers and polynomials and we give some explicit formulae for these numbers and polynomials.
Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P(x_1,...,x_n) are eigenfunctions of a one-parameter family of integral operators Q_z. The operators…
In this article we will derive a combinatorial formula for the partition function p(n). In the second part of the paper we will establish connection between partitions and q-binomial coefficients and give new interpretation for q-binomial…
We study the partially asymmetric exclusion process with open boundaries. We generalise the matrix approach previously used to solve the special case of total asymmetry and derive exact expressions for the partition sum and currents valid…
We consider GL_n(F_q)-analogues of certain factorization problems in the symmetric group S_n: rather than counting factorizations of the long cycle (1, 2, ..., n) given the number of cycles of each factor, we count factorizations of a…
Let $X$ be an algebraic variety over a finite field $\bF_q$, homogeneous under a linear algebraic group. We show that the number of rational points of $X$ over $\bF_{q^n}$ is a periodic polynomial function of $q^n$ with integer…
Variants of the q-hypergeometric equation were introduced in our previous paper with Hatano. In this paper, we consider degenerations of the variant of the q-hypergeometric equation, which is a q-analogue of confluence of singularities in…