4,548 research outputs found
On The Properties Of -Bernstein-Type Polynomials
The aim of this paper is to give a new approach to modified -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
-Bernoulli numbers and -Euler numbers applying -adic -integral
representation on and -adic fermionic -invariant integral
on , respectively, to the inverse of -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
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