Generalized <Emphasis Type="Italic">q</Emphasis>-Baskakov operators

In the present paper we propose a generalization of the Baskakov operators, based on q integers. We also estimate the rate of convergence in the weighted norm. In the last section, we study some shape preserving properties and the property of monotonicity of q-Baskakov operators.     

On discrete <Emphasis Type="Italic">q</Emphasis>-beta operators

The aim of the present paper is to introduce and study the q-analogue of discrete beta operators. First, we show some approximation properties of these operators. Then, we establish some global direct error estimates for the above operators using the second order Ditzian–Totik modulus of smoothness. Finally, we define and study the limit discrete q-beta operator.     

Some <Emphasis Type="Italic">q</Emphasis>-congruences for homogeneous and quasi-homogeneous multiple <Emphasis Type="Italic">q</Emphasis>-harmonic sums

We show some new Wolstenholme type q-congruences for some classes of multiple q-harmonic sums of arbitrary depth with strings of indices composed of ones, twos, and threes. Most of these results are q-extensions of the corresponding congruences for ordinary multiple harmonic sums obtained by the authors in a previous paper. We also establish duality congruences for multiple q-harmonic non-strict sums and a kind of duality for multiple q-harmonic strict sums. Finally, we pose a conjecture concerning two kinds of cyclic sums of multiple q-harmonic sums.     

The moments for <Emphasis Type="Italic">q</Emphasis>-Bernstein operators in the case 0 < <Emphasis Type="Italic">q</Emphasis> < 1

In this note we give the estimates of the central moments for q-Bernstein operators (0 < q < 1) which can be used for studying the approximation properties of the operators.     

A study of (<Emphasis Type="Italic">x</Emphasis>(<Emphasis Type="Italic">q</Emphasis> + 1), <Emphasis Type="Italic">x</Emphasis>; 2, <Emphasis Type="Italic">q</Emphasis>)-minihypers

In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x ≤ q ² − q, not decomposable in the sum of another minihyper and a line, a (j(q + 1), j; 2, q)-minihyper, where j = q ² − q − x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward (Des Codes Cryptogr 44:169–196, 2007), giving further results on these minihypers.     

Approximation by <Emphasis Type="Italic">q</Emphasis>-Bernstein type operators

Using the q-Bernstein basis, we construct a new sequence {L _n } of positive linear operators in C[0, 1]. We study its approximation properties and the rate of convergence in terms of modulus of continuity.     

Approximation by <Emphasis Type="Italic">q</Emphasis> Baskakov Beta operators

In the present paper we introduce the q analogue of the Baskakov Beta operators.We establish some direct results in the polynomial weighted space of continuous functions defined on the interval [0,∞) .Then we obtain point-wise estimate,using the Lipschitz type maximal function.     

The tree method for multidimensional <Emphasis Type="Italic">q</Emphasis>-Hahn and <Emphasis Type="Italic">q</Emphasis>-Racah polynomials

We develop a tree method for multidimensional q-Hahn polynomials. We define them as eigenfunctions of a multidimensional q-difference operator and we use the factorization of this operator as a key tool. Then we define multidimensional q-Racah polynomials as the connection coefficients between different bases of q-Hahn polynomials. We show that our multidimensional q-Racah polynomials may be expressed as product of ordinary one-dimensional q-Racah polynomial by means of a suitable sequence of transplantations of edges of the trees. Our paper is inspired to the classical tree methods in the theory of Clebsch–Gordan coefficients and of hyperspherical coordinates. It is based on previous work of Dunkl, who considered two-dimensional q-Hahn polynomials. It is also related to a recent paper of Gasper and Rahman: we show that their multidimensional q-Racah polynomials correspond to a particular case of our construction.     

On the <Emphasis Type="Italic">k</Emphasis>-gamma <Emphasis Type="Italic">q</Emphasis>-distribution

We provide combinatorial as well as probabilistic interpretations for the q-analogue of the Pochhammer k-symbol introduced by Díaz and Teruel. We introduce q-analogues of the Mellin transform in order to study the q-analogue of the k-gamma distribution.     

Charakterisierung der <Emphasis Type="Italic">q</Emphasis>-Orthogonalpolynome in <Emphasis Type="Italic">x</Emphasis>

Characterization of q-Orthogonal Polynomials. Im Anschluß an die Arbeit Orthogonalpolynome in x und q^–x als Lösungen von reellen q-Operatorgleichungen zweiter Ordnung (Monatsh. Math. 132, 123–140 (2001); im folgenden als [4] zitiert) werden alle Möglichkeiten für q-Orthogonalpolynome in x als Lösungen von q-Operatorgleichungen zweiter Ordnung angegeben (Orthogonalität im positiv definiten Sinne). Dabei erfolgt die Numerierung der Abschnitte und die Angabe der Formel-nummern unter Einbeziehung von [4].     

Multiplicities of simple modules in the Sp(4, <Emphasis Type="Italic">q</Emphasis>)-permutation module on <Emphasis Type="Italic">P</Emphasis>(3, <Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">q</Emphasis> even

In this paper we shall determine the multiplicities of simple modules in characteristic 2 in the Sp(4, q)-permutation module on projective 3-space P(3, q), q = 2 ⁿ .     

On a <Emphasis Type="Italic">q</Emphasis>-analogue of Stancu operators

This paper is concerned with a generalization in q-Calculus of Stancu operators. Involving modulus of continuity and Lipschitz type maximal function, we give estimates for the rate of convergence. A probabilistic approach is presented and approximation properties are established.     

On the directions problem in <Emphasis Type="Italic">AG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We prove that if q = p ^h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q ^k and 1 < k < n, determining N directions where

<Emphasis Type="Italic">q</Emphasis>-Frequent hypercyclicity in spaces of operators

We provide conditions for a linear map of the form \(C_{R,T}(S)=RST\) to be q-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if R is a bounded operator satisfying the q-frequent hypercyclicity criterion, then the map \(C_{R}(S)=RSR^*\) is shown to be q-frequently hypercyclic on the space \(\mathcal {K}(H)\) of all compact operators and the real topological vector space \(\mathcal {S}(H)\) of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for \(C_{R,T}\) to be q-frequently hypercyclic on the Schatten von Neumann classes \(S_p(H)\). We also characterize frequent hypercyclicity of \(C_{M^*_\varphi ,M_\psi }\) on the trace-class of the Hardy space, where the symbol \(M_\varphi \) denotes the multiplication operator associated to \(\varphi \).     

A Characterization of PSL(3, <Emphasis Type="Italic">q</Emphasis>) for <Emphasis Type="Italic">q</Emphasis>=2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>

The order components of a finite group are introduced in [12]. In [9], it is proved that the group PSL(3,q), where q is an odd prime power, is uniquely determined by its order components. In this paper, we show that the group PSL(3, q), where q=2 ^m , is also uniquely determined by its order components. Received December 15, 2000, Revised August 15, 2001, Accepted November 13, 2001     

Some congruences on <Emphasis Type="Italic">q</Emphasis>-Catalan numbers

We establish several q-analogues of congruences involving Catalan numbers. Some of these extend the results of Z.-W. Sun.     

Some new expansions for certain truncated <Emphasis Type="Italic">q</Emphasis>-series

We introduce the notion of \(\mathcal {R}_{\mu }\)-classical orthogonal polynomials, where \(\mathcal {R}_{\mu }\) is the degree raising shift operator for the sequence of Laguerre polynomials of parameter \(\mu \). Then we show that the Laguerre polynomials \(L^{(\mu )}_n(x), \ \mu \ne -m, \ m\ge 0\), are the only \(\mathcal {R}_{\mu }\)-classical orthogonal polynomials.     

Some Asymptotic Results on <Emphasis Type="Italic">q</Emphasis>-Binomial Coefficients

We look at the asymptotic behavior of the coefficients of the q-binomial coefficients (or Gaussian polynomials) \({\left(\begin{array}{c}{a+k}\\{k}\end{array}\right)_{q}}\), when k is fixed. We give a number of results in this direction, some of which involve Eulerian polynomials and their generalizations.     

Smooth noncompact operators from <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>), <Emphasis Type="Italic">K</Emphasis> scattered

Let X be a Banach space, K be a scattered compact and T: B _C(K) → X be a Fréchet smooth operator whose derivative is uniformly continuous. We introduce the smooth biconjugate T**: B _C(K)** → X** and prove that if T is noncompact, then the derivative of T** at some point is a noncompact linear operator. Using this we conclude, among other things, that either is compact or that ℓ₁ is a complemented subspace of X*. We also give some relevant examples of smooth functions and operators, in particular, a C ^1,u -smooth noncompact operator from B _{c O} which does not fix any (affine) basic sequence. P. Hájek was supported by grants A100190502, Institutional Research Plan AV0Z10190503.