In this paper we construct the new analogues of Genocchi the numbers and polynomials. We also observe the behavior of complex roots of the q-Genocchi polynomials Gn,q x , using numerical investigation. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the q-Genocchi polynomials Gn,q x . Finally, we give a table for the solutions of the q-Ge...

We introduce and study a one-parameter generalization of the q –Whittaker symmetric functions. This is family multivariate polynomials, whose construction may be viewed as an application procedure fusion from integrable lattice models to vertex model interpretation Hall–Littlewood polynomials [3] , [6] [7] . prove branching Pieri rules, standard dual (skew) Cauchy summation identities, integral...

We show that limit transitions from Askey-Wilson polynomials to q-Racah, little and big q-Jacobi polynomials can be made rigorous on the level of their orthogonality measures in a suitable weak sense. This allows us to derive the orthogonality relations and norm evaluations for the q-Racah polynomials, little and big q-Jacobi polynomials by taking limits in the orthogonality relations and norm ...

Knop and Sahi simultaneously introduced a family of non-homogeneous, non-symmetric polynomials, Gα(x; q, t). The top homogeneous components of these polynomials are the non-symmetric Macdonald polynomials, Eα(x; q, t). An appropriate Hecke algebra symmetrization of Eα yields the Macdonald polynomials, Pλ(x; q, t). A search for explicit formulas for the polynomials Gα(x; q, t) led to the main re...

Applying Baxter’s method of the Q-operator to the set of Sekiguchi’s commuting partial differential operators we show that Jack’s symmetric polynomials P (1/g) λ (x1, . . . , xn) are eigenfunctions of a one-parameter family of integral operators Qz . The operators Qz are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Qzk we construct...

In this paper, we discuss the properties of q polynomials which are hold in any way, especially difference-differential equation and similar relations. Also, the distribution of zeros is studied. At last, we illustrate all by a few examples and make some conjectures.