A curious q-analogue of Hermite polynomials

Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider a new family of q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci and q-Lucas polynomials. The latter relation yields a generalization of the Touchard-Riordan formula.     

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.     

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.     

A new q-analogue of the sum of cubes

A new q-analogue of the sum of cubes is given with a combinatorial interpretation on the lattice of subspaces.     

Approximation properties for generalized q-Bernstein polynomials

In this paper, we introduce the generalized q-Bernstein polynomials based on the q-integers and we study approximation properties of these operators. In special case, we obtain Stancu operators or Phillips polynomials.     

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.     

Some properties of q-biorthogonal polynomials

Almost four decades ago, Konhauser introduced and studied a pair of biorthogonal polynomials

     

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.     

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.     

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 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.     

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.     

Limit relations for the complex zeros of Laguerre and q-Laguerre polynomials

For each the nth Laguerre polynomial has an m-fold zero at the origin when α=−m. As the real variable α→−m, it has m simple complex zeros which approach 0 in a symmetric way. This symmetry leads to a finite value for the limit of the sum of the reciprocals of these zeros. There is a similar property for the zeros of the q-Laguerre polynomials and of the Jacobi polynomials and similar results hold for sums of other negative integer powers.     

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.     

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 p-adic q-l-functions and sums of powers

In this paper, we give an explicit p-adic expansion of

     

Second structure relation for q-semiclassical polynomials of the Hahn Tableau

The q-classical orthogonal polynomials of the q-Hahn Tableau are characterized from their orthogonality condition and by a first and a second structure relation. Unfortunately, for the q-semiclassical orthogonal polynomials (a generalization of the classical ones) we find only in the literature the first structure relation. In this paper, a second structure relation is deduced. In particular, by means of a general finite-type relation between a q-semiclassical polynomial sequence and the sequence of its q-differences such a structure relation is obtained.     

Extensions of q-Chu-Vandermonde's identity

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