Two new q-exponential operator identities and their applications

In this paper, we use the q-Chu–Vandermonde formula to prove two new operator identities, which are the extensions of Liu's results. These two q-exponential operator identities are used to derive some q-summation formulas and q-integrals.     

Extensions of q-Chu-Vandermonde's identity

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

A family of p-adic twisted interpolation functions associated with the modified Bernoulli numbers

The main purpose of this paper is to construct a family of modified p-adic twisted functions, which interpolate the modified twisted q-Bernoulli polynomials and the generalized twisted q-Bernoulli numbers at negative integers. We also give some applications and examples related to these functions and numbers.     

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.     

Applications of the Mellin transform in quantum calculus

In this paper, we study in quantum calculus the correspondence between poles of the q-Mellin transform (see [A. Fitouhi, N. Bettaibi, K. Brahim, The Mellin transform in Quantum Calculus, Constr. Approx. 23 (3) (2006) 305-323]) and the asymptotic behaviour of the original function at 0 and ∞. As applications, we give a new technique (in q-analysis) to derive the asymptotic expansion of some functions defined by q-integrals or by q-harmonic sums. Finally, a q-analogue of the Mellin-Perron formula is given.     

Arithmetic properties of q-Fibonacci numbers and q-pell numbers

We investigate some arithmetic properties of the q-Fibonacci numbers and the q-Pell numbers.     

A note on the modified q-Bernstein polynomials for functions of several variables and their reflections on q-Volkenborn integration

In this paper, we consider the modified q-Bernstein polynomials for functions of several variables on q-Volkenborn integral and investigate some new interesting properties of these polynomials related to q-Stirling numbers, Hermite polynomials and Carlitz’s type q-Bernoulli numbers.     

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.     

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.     

On rubin's harmonic analysis and its related positive definite functions

In this paper, a new formulation of the Rubin's q-translation is given, which leads to a reliable q-harmonic analysis. Next, related q-positive definite functions are introduced and studied, and a Bochner's theorem is proved.     

Factors of the Gaussian coefficients

We present some simple observations on factors of the q-binomial coefficients, the q-Catalan numbers, and the q-multinomial coefficients. Writing the Gaussian coefficient with numerator n and denominator k in a form such that 2k?n by the symmetry in k, we show that this coefficient has at least k factors. Some divisibility results of Andrews, Brunetti and Del Lungo are also discussed.     

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 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.     

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 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.     

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.     

q-Dedekind type sums related to q-zeta function and basic L-series

The main purpose of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Riemann zeta function, q-analogue Hurwitz zeta function, q-analogue Dirichlet L-function and two-variable q-L-function. In particular, by using these generating functions, we will construct new generating functions which produce q-Dedekind type sums and q-Dedekind type sums attached to Dirichlet character. We also give the relations between these sums and Dedekind sums. Furthermore, by using *-product which is given in this paper, we will give the relation between Dedekind sums and q-L function as well.