Related papers: An Operator Approach to the Al-Salam-Carlitz Polyn…

We study resolvent estimates for non-selfadjoint semiclassical pseudodifferential operators with double characteristics. Assuming that the quadratic approximation along the double characteristics is elliptic, we obtain polynomial upper…

We derive and discuss a technique for manipulating power series which is complementary to standard procedures. We begin with the translation operator, but we express the operator as an infinite product instead of expanding it as a series…

In this paper we study that the $q$-Euler numbers and polynomials are analytically continued to $E_q(s)$. A new formula for the Euler's $q$-Zeta function $\zeta_{E,q}(s)$ in terms of nested series of $\zeta_{E,q}(n)$ is derived. Finally we…

A well-known characterization of Jordan vectors of a matrix polynomial $L(z)$ is generalized to a characterization of Jordan vectors of the operator-valued function $Q(z)$ at an eigenvalue $\alpha \in \mathbb{C}$. The results are then…

In this paper, we deal with q-Euler numbers and q-Bernoulli numbers. We derive some interesting relations for q-Euler numbers and polynomials by using their generating function and derivative operator. Also, we show between the q-Euler…

Using a realization of the q-exponential function as an infinite multiplicative sereis of the ordinary exponential functions we obtain new nonlinear connection formulae of the q-orthogonal polynomials such as q-Hermite, q-Laguerre and…

Using notions of composita and composition of generating functions we obtain explicit formulas for Chebyshev polynomials, Legendre polynomials, Gegenbauer polynomials, Associated Laguerre polynomials, Stirling polynomials, Abel polynomials,…

Let $f=\sum_{n=0}^\infty f_n x^n \in \overline{\mathbb Q}[[x]$ be a solution of an algebraic differential equation $Q(x,y(x), \ldots, y^{(k)}(x))=0$, where $Q$ is a multivariate polynomial with coefficients in $\overline{\mathbb Q}$. The…

In this paper, we introduce the concept of the (higher order) Appell-Carlitz numbers which unifies the definitions of several special numbers in positive characteristic, such as the Bernoulli-Carlitz numbers and the Cauchy-Carlitz…

In this paper, we study multiplication formula of $F$-polynomial of representations of Hernandez and Leclerc's quivers with potentials. Since the truncated $q$-characters of some real simple modules over a quantum affine group…

Recently, Mursaleen et al applied (p,q)-calculus in approximation theory and introduced (p,q)-analogue of Bernstein operators in [16]. In this paper, we construct and introduce a generalization of the bivariate Bleimann-Butzer-Hahn…

In this paper, we present some criteria for the $2$-$q$-log-convexity and $3$-$q$-log-convexity of combinatorial sequences, which can be regarded as the first column of certain infinite triangular array $[A_{n,k}(q)]_{n,k\geq0}$ of…

The purpose of this article is to give a Chlodowsky type generalization of Szasz operators defined by means of the Sheffer type polynomials. We obtain convergence properties of our operators with the help of Korovkin's theorem and the order…

The n-point correlation functions introduced by Bloch and Okounkov have already found several geometric connections and algebraic generalizations. In this Note we formulate a q,t-deformation of this n-point function. The key operator used…

Using the method of the $q$-exponential differential operator, we give an extension of the Sears $_4\phi_3$ transformation formula. Based on this extended formula and a $q$-series expansion formula for an analytic function around the…

Given a set of matrices, modeled as samples of a matrix-valued function, we suggest a method to approximate the underline function using a product approximation operator. This operator extends known approximation methods by exploiting the…

In this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of…

We develop and compare two algorithms for computing first-order right-hand factors in the ring of linear Mahler operators$\ell_r M^r + \dots + \ell_1 M + \ell_0$where $\ell_0, \dots, \ell_r$ are polynomials in~$x$ and $Mx = x^b M$ for some…

We provide an exact regular solution of an operator system arising as the prolongation structure associated with the heavenly equation. This solution is expressed in terms of operator Bessel coefficients.

We characterize all linear operators on finite or infinite-dimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and non-negative polynomials, respectively. This is done by means of Fischer-Fock…