Extensions of q-Chu-Vandermonde's identity

In this paper, we apply q-exponential operator to get some general q-Chu-Vandermonde's identities.     

Two operator identities and their applications to terminating basic hypergeometric series and q-integrals

In this paper, we first give two interesting operator identities, and then, using them and the q-exponential operator technique to some terminating summation formulas of basic hypergeometric series and q-integrals, we obtain some q-series identities and q-integrals involving ₃?₂.     

A note on generalized q-difference equations for q-beta and Andrews–Askey integral

Two q-difference equations with solutions expressed by q-exponential operator identities are investigated. As applications, two extensions of Ramanujan?s formulas for q-beta integral are given, two generalizations of Andrews–Askey integral are obtained. In addition, generating functions for generalized Al-Salam–Carlitz polynomials are deduced. At last, a generalized transformation identity is gained.     

Another homogeneous q-difference operator

In this paper we show the equivalence between Goldman-Rota q-binomial identity and its inverse. We may specialize the value of the parameters in the generating functions of Rogers-Szegö polynomials to obtain some classical results such as Euler identities and the relation between classical and homogeneous Rogers-Szegö polynomials. We give a new formula for the homogeneous Rogers-Szegö polynomials h_n(x,y|q). We introduce a q-difference operator θ_xy on functions in two variables which turn out to be suitable for dealing with the homogeneous form of the q-binomial identity. By using this operator, we got the identity obtained by Chen et al. [W.Y.C. Chen, A.M. Fu, B. Zhang, The homogeneous q-difference operator, Advances in Applied Mathematics 31 (2003) 659-668, Eq. (2.10)] which they used it to derive many important identities. We also obtain the q-Leibniz formula for this operator. Finally, we introduce a new polynomials s_n(x,y;b|q) and derive their generating function by using the new homogeneous q-shift operator L(bθ_xy).     

q-Difference equation and the Cauchy operator identities

In this paper, we verify the Cauchy operator identities by a new method. And by using the Cauchy operator identities, we obtain a generating function for Rogers-Szegö polynomials. Applying the technique of parameter augmentation to two multiple generalizations of q-Chu-Vandermonde summation theorem given by Milne, we also obtain two multiple generalizations of the Kalnins-Miller transformation.     

Inversion formulas for q-Riemann-Liouville and q-Weyl transforms

We study fractional transforms associated with q-Bessel operator which is useful to inverse q-Riemann-Liouville and q-Weyl transforms.     

A q-summation formula,the continuous q-Hahn polynomials and the big q-Jacobi polynomials

Using a general q-summation formula, we derive a generating function for the q-Hahn polynomials, which is used to give a complete proof of the orthogonality relation for the continuous q-Hahn polynomials. A new proof of the orthogonality relation for the big q-Jacobi polynomials is also given. A simple evaluation of the Nassrallah–Rahman integral is derived by using this summation formula. A new q-beta integral formula is established, which includes the Nassrallah–Rahman integral as a special case. The q-summation formula also allows us to recover several strange q-series identities.     

Basic Fourier series on a q-linear grid: Convergence theorems

For 0<q<1 define the symmetric q-linear operator acting on a suitable function f(x) by δf(x)=f(q1/2x)−f(q−1/2x). The q-linear initial value problem , f(0)=1, has two entire functions Cq(z) and Sq(z) as linearly independent solutions, which are orthogonal on a discrete set. Sufficient conditions for pointwise convergence and for uniform convergence of the corresponding Fourier expansion are given.     

New orthogonality relations for the continuous and the discrete q-ultraspherical polynomials

In a recent contribution [N.M. Atakishiyev, A.U. Klimyk, On discrete q-ultraspherical polynomials and their duals, J. Math. Anal. Appl. 306 (2005) 637-645], the so-named discrete q-ultraspherical polynomials were introduced as a specialization of the big q-Jacobi polynomials, and their orthogonality established for values of the parameter outside its commonly known domain but inside the range of validity of the conditions of Favard's theorem. In this paper we consider both the continuous and the discrete q-ultraspherical polynomials and we prove that their orthogonality is guaranteed for the whole range of the allowed parameters, even in those intriguing cases in which the three term recurrence relation breaks down. The presence of either the Askey-Wilson divided difference operator (in the continuous case), or the q-derivative operator (in the discrete one), provides the q-Sobolev character of the non-standard inner products introduced in our approach.     

q-Differential operator identities and applications

In this paper, we construct a new q-exponential operator and obtain some operator identities. Using these operator identities, we give a formal extension of Jackson's transformation formula. A formal extension of Bailey's summation and an extension of the Sears terminating balanced transformation formula are also derived by our operator method. In addition, we also derive several interesting a formal extensions involving multiple sum about three terms of Sears transformation formula and Heine's transformation formula.     

A family of q-Dyson style constant term identities

By generalizing Gessel-Xin's Laurent series method for proving the Zeilberger-Bressoud q-Dyson Theorem, we establish a family of q-Dyson style constant term identities. These identities give explicit formulas for certain coefficients of the q-Dyson product, including three conjectures of Sills' as special cases and generalizing Stembridge's first layer formulas for characters of SL(n,C).     

Perturbation of the q-numerical radius of a weighted shift operator

We formulate the Taylor series expansion for the q-numerical radius of a weighted shift operator with periodic weights near q=0. Coefficients up to the fourth order in the expansion are found via the perturbation theory of Hermitian matrices.     

q-positive definiteness and related operators

We are in progress of extending the family of ‘q-deformed operators’ considered in the previous papers by joining to them q-subnormal as well as q-formally subnormal ones. It turns out that q-positive definiteness, a notion generalizing Halmos' standard positive definiteness of bounded subnormal operators, is likewise central for our new scheme.     

Sobolev type spaces in quantum calculus

In this paper q-Sobolev type spaces are defined on Rq by using the q-cosine Fourier transform and its inverse. In particular, embedding results for these spaces are established. Next we define the q-cosine potential and study some of its properties.     

Some results for the q-Bernoulli and q-Euler polynomials

We show some results for the q-Bernoulli and q-Euler polynomials. The formulas in series of the Carlitz's q-Stirling numbers of the second kind are also considered. The q-analogues of well-known formulas are derived from these results.     

q-Taylor and interpolation series for Jackson q-difference operators

We prove q-Taylor series for Jackson q-difference operators. Absolute and uniform convergence to the original function are proved for analytic functions. We derive interpolation results for entire functions of q-exponential growth which is less than lnq⁻¹, 0<q<1, from its values at the nodes , a is a non-zero complex number with absolute and uniform convergence criteria.     

On the Gaussian q-distribution

We present a study of the Gaussian q-measure introduced by Díaz and Teruel from a probabilistic and from a combinatorial viewpoint. A main motivation for the introduction of the Gaussian q-measure is that its moments are exactly the q-analogues of the double factorial numbers. We show that the Gaussian q-measure interpolates between the uniform measure on the interval [−1,1] and the Gaussian measure on the real line.     

On discrete q-ultraspherical polynomials and their duals

A special case of the big q-Jacobi polynomials Pn(x;a,b,c;q), which corresponds to a=b=−c, is shown to satisfy a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a<q⁻¹). Since Pn(x;qα,qα,−qα;q) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q→1, this family represents another q-extension of these classical polynomials, different from the continuous q-ultraspherical polynomials of Rogers. For a dual family with respect to the polynomials Pn(x;a,a,−a;q) (i.e., for dual discrete q-ultraspherical polynomials) we also find new orthogonality relations with extremal measures.     

On the rate of approximation by q modified Beta operators

In the present paper we propose the q analogue of the modified Beta operators. We apply q-derivatives to obtain the central moments of the discrete q-Beta operators. A direct result in terms of modulus of continuity for the q operators is also established. We have also used the properties of q integral to establish the recurrence formula for the moments of q analogue of the modified Beta operators. We also establish an asymptotic formula. In the end we have also present the modification of such q operators so as to have better estimate.     

Elements of harmonic analysis related to the third basic zero order Bessel function

This paper is devoted to the study of some q-harmonic analysis related to the third q-Bessel function of order zero. We establish a product formula leading to a q-translation with some positive kernel. As an application, we provide a q-analogue of the continuous wavelet transform related to this harmonic analysis.