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.     

Structure of the zeros ofq-Bernoulli polynomials

In this paper we observe the structure of the roots ofq-Bernoulli polynomials,β _n (w, h|q), using numerical investigation. By numerical experiments, we demonstrate a remarkably regular structure of the real roots ofβ _n (w, h|q) forq=?1/5, ?1/2. Finally, we give a table for numbers of real and complex zeros ofβ _n (w, h|q).     

Linearization of the product of symmetric orthogonal polynomials

A new constructive approach is given to the linearization formulas of symmetric orthogonal polynomials. We use the monic three-term recurrence relation of an orthogonal polynomial system to set up a partial difference equation problem for the product of two polynomials and solve it in terms of the initial data. To this end, an auxiliary function of four integer variables is introduced, which may be seen as a discrete analogue of Riemann's function. As an application, we derive the linearization formulas for the associated Hermite polynomials and for their continuousq-analogues. The linearization coefficients are represented here in terms of₃ F ₂ and₃Φ₂ (basic) hypergeometric functions, respectively. We also give some partial results in the case of the associated continuousq-ultraspherical polynomials.     

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).     

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-Bernoulli polynomials and q-umbral calculus

In this paper, we investigate some properties of q-Bernoulli polynomials arising from q-umbral calculus. We find a formula for expressing any polynomial as a linear combination of q-Bernoulli polynomials with explicit coefficients. Also, we establish some connections between q-Bernoulli polynomials and higher-order q-Bernoulli polynomials.     

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.     

Some limit relations in the A1 tableau of Dunkl-Cherednik operators

In this paper we study some limit relations involving some q-special functions related with the A₁ (root system) tableau of Dunkl-Cherednik operators. Concretely we consider the limits involving the nonsymmetric q-ultraspherical polynomials (q-Rogers polynomials), ultraspherical polynomials (Gegenbauer polynomials), q-Hermite and Hermite polynomials.     

The norm ratio of the polynomials with coefficients as binary sequence

Given a positive integerq, the ratio of the 2q-norm of a polynomial which its coefficients form a binary sequence and its 2-norm arose from telecommunication engineering consists of finding any type of such polynomials having the ratio “small”. In this paper we consider some special types of these polynomials, discuss the sharpest possible upper bound, and prove a result for the ratio. MAIN FACTS: A conjecture over a Rudin-Shapiro polynomialP _n which has degree 2 ⁿ ?1 is that for any integerq, the ratio of its 2 _q norm and its 2 norm is asymptotic to the 2qth root of 2 ^q (q+1)^?1. In other words $||P_n ||_{2q} \sim ||P_n ||_2 \sqrt[{2q}]{{\frac{{2q}}{{q + 1}}}}$ . So far only up toq= 2 has been verified. However if the asymptotic behavior is valid for an evenq, then it is also valid for its next consecutive odd integer.     

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.     

A Combinatorial Formula for the Schur Coefficients of the Integral Form of the Macdonald Polynomials in the Two Column and Certain Hook Cases

In this paper, we prove that the integral form of Macdonald polynomials J _?? [X; q, t] has the property that J _?? [X; q, t]/(1?q) ⁿ has Schur expansion with positive polynomial coefficient. Our proof proceeds by constructing constructing combinatorial formula for the Schur coefficients when??? is either a two column shape or a certain type of hook shape.     

Plancherel-Rotach asymptotics for certain basic hypergeometric series

In this work we study the chaotic and periodic asymptotics for the confluent basic hypergeometric series. For a fixed q∈(0,1), the asymptotics for Euler's q-exponential, q-Gamma function Γq(x), q-Airy function of K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, Ramanujan function (q-Airy function), Jackson's q-Bessel function of second kind, Ismail-Masson orthogonal polynomials (q⁻¹-Hermite polynomials), Stieltjes-Wigert polynomials, q-Laguerre polynomials could be derived as special cases.     

U(n) Wigner coefficients,the path sum formula,and invariant G-functions

We prove the path sum formula for computing the U(n) invariant denominator functions associated to stretched U(n) Wigner operators. A family of U(n) invariant polynomials G_[λ]⁽ⁿ⁾ is then defined which generalize the _μG_q⁽ⁿ⁾ polynomials previously studied. The G_[λ]⁽ⁿ⁾ polynomials are shown to satisfy a number of difference equations and have symmetry properties similar to the _μG_q⁽ⁿ⁾ polynomials. We also give a direct proof of the important transposition symmetry for the G_[λ]⁽ⁿ⁾ polynomials. To enable the non-specialist to understand the foundations for these remarkable polynomials, we provide an exposition of the boson calculus and the construction of the multiplicity-free U(n) Wigner operators.     

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.     

Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials

In the spaceL _q, 1<q<∞ we establish estimates for the orders of the best approximations of the classes of functions of many variablesB _1,θ ^r andB _p,α ^r by orthogonal projections of functions from these classes onto the subspaces of trigonometric polynomials. It is shown that, in many cases, the estimates obtained in the present work are better in order than in the case of approximation by polynomials with harmonics from the hyperbolic cross.     

A generalization of the Syracuse algorithm in Fq[x]

In this note we remark that while much of the theory of a recent paper of Matthews and Watts on mappings T: Z → Z generalizing the Syracuse algorithm also goes over to mappings T: F_q[x] → F_q[x], the conjectural picture is not as clear for polynomials. We exhibit two divergent trajectories which possess an unexpected regularity, and which do not obey a certain expected uniformity of distribution.     

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.     

Symmetric Polynomials and Some Good Codes

Elements from extensions of F_q are employed to construct a class of linear codes over F_q with good parameters through symmetric polynomials over F_q.     

On some properties of q-Hahn multiple orthogonal polynomials

This contribution deals with multiple orthogonal polynomials of type II with respect to q-discrete measures (q-Hahn measures). In addition, we show that this family of multiple orthogonal polynomials has a lowering operator, and raising operators, as well as a Rodrigues type formula. The combination of lowering and raising operators leads to a third order q-difference equation when two orthogonality conditions are considered. An explicit expression of this q-difference equation will be given. Indeed, this q-difference equation relates polynomials with a given degree evaluated at four consecutive non-uniformed distributed points, which makes these polynomials interesting from the point of view of bispectral problems.