q-Hardy-Berndt type sums associated with q-Genocchi type zeta and q-l-functions

The aim of this paper is to define new generating functions. By applying a derivative operator and the Mellin transformation to these generating functions, we define q-analogue of the Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, and q-Genocchi type l-function. We define partial zeta function. By using this function, we construct p-adic interpolation functions which interpolate generalized q-Genocchi numbers at negative integers. We also define p-adic meromorphic functions on C_p. Furthermore, we construct new generating functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations, related to these sums.     

Generalized discrete q-Hermite I polynomials and q-deformed oscillator

In this paper, we present an explicit realization of q-deformed Calogero-Vasiliev algebra whose generators are first-order q-difference operators related to the generalized discrete q-Hermite I polynomials recently introduced in [14]. Furthermore, we construct the wave functions and we determine the q-coherent states.     

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.     

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.     

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.     

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.     

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.     

On asymptotics of the q-exponential and q-gamma functions

In this work we present a derivation for the complete asymptotic expansions of Euler?s q-exponential function and Jackson?s q-gamma function via Mellin transform. These formulas are valid everywhere, uniformly on any compact subset of the complex plane.     

Combinatorial interpretations of the q-Faulhaber and q-Salié coefficients

Recently, Guo and Zeng discovered two families of polynomials featuring in a q-analogue of Faulhaber's formula for the sums of powers and a q-analogue of Gessel-Viennot's formula involving Salié's coefficients for the alternating sums of powers. In this paper, we show that these are polynomials with symmetric, nonnegative integral coefficients by refining Gessel-Viennot's combinatorial interpretations.     

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.     

On the twisted q-Euler numbers and polynomials associated with basic q-l-functions

One of the purposes of this paper is to construct the twisted q-Euler numbers by using p-adic invariant integral on Zp in the fermionic sense. Moreover, we consider the twisted Euler q-zeta functions and q-l-functions which interpolate the twisted q-Euler numbers and polynomials at a negative integer.     

q-Partition algebra combinatorics

We study a q-analog Qr(n,q) of the partition algebra Pr(n). The algebra Qr(n,q) arises as the centralizer algebra of the finite general linear group GLn(Fq) acting on a vector space coming from r-iterations of Harish-Chandra restriction and induction. For n?2r, we show that Qr(n,q) has the same semisimple matrix structure as Pr(n). We compute the dimension to be a q-polynomial that specializes as dn,r(1)=nr and dn,r(0)=B(r), the rth Bell number. Our method is to write dn,r(q) as a sum over integer sequences which are q-weighted by inverse major index. We then find a basis of indexed by n-restricted q-set partitions of {1,…,r} and show that there are dn,r(q) of these.     

The q-Calkin-Wilf tree

We define a q-analogue of the Calkin-Wilf tree and the Calkin-Wilf sequence. We show that the nth term f(n;q) of the q-analogue of the Calkin-Wilf sequence is the generating function for the number of hyperbinary expansions of n according to the number of powers that are used exactly twice. We also present formulae for branches within the q-analogue of the Calkin-Wilf tree and predecessors and successors of terms in the q-analogue of the Calkin-Wilf sequence.     

Extensions of q-Chu-Vandermonde's identity

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

On q-orthogonal polynomials, dual to little and big q-Jacobi polynomials

We derive discrete orthogonality relations for polynomials, dual to little and big q-Jacobi polynomials. This derivation essentially requires use of bases, consisting of eigenvectors of certain self-adjoint operators, which are representable by a Jacobi matrix. Recurrence relations for these polynomials are also given.     

On a q-Paley-Wiener theorem

After the analysis of the q-even translation and the q-cosine Fourier transform by A. Fitouhi and F. Bouzeffour in [q-cosine Fourier transform and q-heat equation, Ramanujan J., in press], it is natural to look for the q-analogue of some well-known classical theorems. In this paper, we purpose to give a q-version of the Paley-Wiener theorem related to this Fourier transform using standard methods of the q-calculus.     

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).     

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-Analog of tableau containment

We prove a q-analog of the following result due to McKay, Morse and Wilf: the probability that a random standard Young tableau of size n contains a fixed standard Young tableau of shape λ?k tends to fλ/k! in the large n limit, where fλ is the number of standard Young tableaux of shape λ. We also consider the probability that a random pair (P,Q) of standard Young tableaux of the same shape contains a fixed pair (A,B) of standard Young tableaux.