On The Properties Of qq-Bernstein-Type Polynomials

The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling numbers and generalized Bernoulli polynomials are derived. Moreover, the generating function, interpolation function of these polynomials of several variables and also the derivatives of these polynomials and their generating function are given. Finally, we get new interesting identities of modified $q$-Bernoulli numbers and $q$-Euler numbers applying $p$-adic $q$-integral representation on $\mathbb {Z}_p$ and $p$-adic fermionic $q$-invariant integral on $\mathbb {Z}_p$, respectively, to the inverse of $q$-Bernstein polynomials.Comment: 17 pages, some theorems added to new versio

Generalized Stirling Permutations and Forests: Higher-Order Eulerian and Ward Numbers

We define a new family of generalized Stirling permutations that can be interpreted in terms of ordered trees and forests. We prove that the number of generalized Stirling permutations with a fixed number of ascents is given by a natural three-parameter generalization of the well-known Eulerian numbers. We give the generating function for this new class of numbers and, in the simplest cases, we find closed formulas for them and the corresponding row polynomials. By using a non-trivial involution our generalized Eulerian numbers can be mapped onto a family of generalized Ward numbers, forming a Riordan inverse pair, for which we also provide a combinatorial interpretation.We are indebted to Alan Sokal for his participation in the early stages of this work, his encouragement, and useful suggestions later on. We also thank Jesper Jacobsen, Anna de Mier, Neil Sloane, and Mike Spivey for correspondence, and David Callan for pointing out some interesting references to us. Last but not least, we thank Bojan Mohar for valuable criticisms and suggestions

A new approach to modified q-Bernstein polynomials for functions of two variables with their generating and interpolation functions

The aim of this paper is to give a new approach to modified q-Bernstein polynomials for functions of two variables. By using these type polynomials, we derive recurrence formulas and some new interesting identities related to the second kind Stirling numbers and generalized Bernoulli polynomials. Moreover, we give the generating function and interpolation function of these modified q-Bernstein polynomials of two variables and also give the derivatives of these polynomials and their generating function.Comment: 11 page

On the families of q-Euler numbers and polynomials and their applications

In the present paper, we investigate special generalized q-Euler numbers and polynomials. Some earlier results of T. Kim in terms of q-Euler polynomials with weight alpha can be deduced. For presentation of our formulas we apply the method of generating function and p-adic q-integral representation on Zp. We summarize our results as follows. In section 2, by using combinatorial techniques we present two formulas for q-Euler numbers with weight alpha. In section 3, we derive distribution formula (Multiplication Theorem) for Dirichlet type of q-Euler numbers and polynomials with weight . Moreover we define partial Dirichlet type zeta function and Dirichlet q-L-function, and obtain some interesting combinatorial identities for interpolating our new definitions. In addition, we derive behavior of the Dirichlet type of q-Euler L-function with weight alpha, Lq (s; x j) at s = 0. Furthermore by using second kind stirling numbers, we obtain an explicit formula for Dirichlet type q-Euler numbers with weight alpha. Moreover a novel formula for q-Euler-Zeta function with weight in terms of nested series of E;q (n j) is derived . In section 4, by introducing p-adic Dirichlet type of q-Euler measure with weight, and we obtain some combinatorial relations, which interpolate our previous results. In section 5, which is the main section of our paper. As an application, we introduce a novel concept of dynamics of the zeros of analytically continued q-Euler polynomials with weight alpha.Comment: 15 pages, submitte

Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem

We consider the numbers arising in the problem of normal ordering of expressions in canonical boson creation and annihilation operators. We treat a general form of a boson string which is shown to be associated with generalizations of Stirling and Bell numbers. The recurrence relations and closed-form expressions (Dobiski-type formulas) are obtained for these quantities by both algebraic and combinatorial methods. By extensive use of methods of combinatorial analysis we prove the equivalence of the aforementioned problem to the enumeration of special families of graphs. This link provides a combinatorial interpretation of the numbers arising in this normal ordering problem.Comment: 10 pages, 5 figure