Several authors have examined connections between restricted permutations and Chebyshev polynomials of the second kind. In this paper we prove analogues of these results for colored permutations. First we define a distinguished set of length two and length three patterns, which contains only 312 when just one color is used. Then we give a recursive procedure for computing the generating functio...

Among the simplest of the classical polynomials are the Chebyshev polynomials of the first and second kind, Tk(x) and Uk(x). In our normalization, the indices are allowed to be halfintegers as well as integers, and the “polynomials” actually live in Z[x, √ 2− x, √ 2 + x]. We will show that the rational functions Tm/2(x) Tn/2(x) and Um/2(x) Un/2(x) are very remarkable from the point of view of G...

We determine which (ordinary) Riordan arrays are the coefficient arrays of a family of orthogonal polynomials. In so doing, we are led to introduce a family of polynomials, which includes the Boubaker polynomials, and a scaled version of the Chebyshev polynomials, using the techniques of Riordan arrays. We classify these polynomials in terms of the Chebyshev polynomials of the first and second ...

In this paper, we study the asymptotics of the discrete Chebyshev polynomials tn(z,N) as the degree grows to infinity. Global asymptotic formulas are obtained as n → ∞, when the ratio of the parameters n/N = c is a constant in the interval (0, 1). Our method is based on a modified version of the Riemann-Hilbert approach first introduced by Deift and Zhou.

This paper presents the boundary integral equation method (BIEM) for stress intensity factors and elasticity T-stresses evaluation in 3D problems. Flat rectangular, elliptic, penny-shaped cracks rectangular crack on a cylindrical surface have been investigated. The hyper-singular integrals are treated with Taylor’s series expansion of kernel, Chebyshev polynomials second kind used to solve equa...

The time-fractional heat equation governed by nonlocal conditions is solved using a novel method developed in this study, which based on the spectral tau method. There are two sets of basis functions used. first set non-symmetric polynomials, namely, shifted Chebyshev polynomials sixth-kind (CPs6), and second modified CPs6. approximation solution written as product chosen function sets. For met...

In some applications one is led to consider perturbations of orthogonal polynomials translated by a modification on the first coefficients of the second order recurrence relation satisfied by these polynomials. Moreover, the four Chebyshev families are among the most useful orthogonal sequences due to their exceptional features. Thus, it is important to clarify and explicit the properties of pe...

The concept of Lucky k-polynomials and in particular Lucky χ-polynomials was recently introduced. This paper introduces Stirling number of the fourth kind and Lucky partitions of a finite set in order to determine either the Lucky k- or Lucky χ-polynomial of a graph. The integer partitions influence Stirling partitions of the second kind.

The main goal of the present paper is to construct some families of the Carlitz's $q$-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz's $q$-Bernoulli polynomials and numbers with weight ($_{p}$. We then define the modified degenerate Carlitz's $q$-Bernoulli polynomials and numbers with weight ($alpha ,beta $) and obtain some recurrence relations and other identities...

We give two recursive expressions for both MacWilliams and Chebyshev matrices. The expressions give rise to simple recursive algorithms for constructing the matrices. In order to derive the second recursion for the Chebyshev matrices we find out the Krawtchouk coefficients of the Discrete Chebyshev polynomials, a task interesting on its own.