The optical behaviour of the (human) cornea is often characterized with the Zernike-coefficients derived via the Zernike-transform of its optical power map. In this paper, a radial transform based on the Chebyshev-polynomials of the second kind is suggested for a surface-based, rather than an optical power map based representation of the cornea. This transform is well-suited for providing compa...

Abstract In this paper, the numerical method for solving Abel’s integral equations is presented. This method is based on fractional calculus. Also, Chebyshev polynomials are utilized to apply fractional properties for solving Abel’s integral equations of the first and second kind. The fractional operator is considered in the sense of RiemannLiouville. Although Abel’s integral equations as singu...

Abstract. A permutation is called layered if it consists of the disjoint union of substrings (layers) so that the entries decrease within each layer, and increase between the layers. We find the generating function for the number of permutations on n letters avoiding (1, 2, 3) and a layered permutation on k letters. In the most interesting case of two layers, the generating function depends onl...

In this work we study the different type of regular boundary value problems on a path associated with the Schrödinger operator. In particular, we obtain the Green function for each problem and we emphasize the case of Sturm-Liouville boundary conditions. In any case, the Green function is given in terms of second kind Chebyshev polynomials since they verify a recurrence law similar to the one v...

We study generating functions for the number of involutions, even involutions, and odd involutions in Sn subject to two restrictions. One restriction is that the involution avoid 3412 or contain 3412 exactly once. The other restriction is that the involution avoid another pattern τ or contain τ exactly once. In many cases we express these generating functions in terms of Chebyshev polynomials o...

In this study, we describe an algorithm that computes the degree of a Dickson Polynomial First Kind from its known value at point. Our is based on mathematical relation between Polynomials and Chebyshev Kind.

Abstract The principal aim of the current article is to establish new formulas Chebyshev polynomials sixth-kind. Two different approaches are followed derive connection between these and some other orthogonal polynomials. coefficients expressed in terms terminating hypergeometric functions certain arguments; however, they can be reduced cases. New moment sixth-kind also established, virtue such...

Orthogonal polynomials have very useful properties in the mathematical problems, so recent years have seen a great deal in the field of approximation theory using orthogonal polynomials. In this paper, we characterize a sequence of the generalized Chebyshev-type polynomials of the first kind { T (M,N) n (x) } n∈N∪{0} , which are orthogonal with respect to the measure √ 1−x2 π dx + Mδ−1 + Nδ1, w...

The properties of two families of s-orthogonal polynomials, which are connected with Chebyshev polynomials of third and fourth kind, are studied. Evaluations of the remainders are given and asymptotic formulae are calculated for the corresponding hyper-Gaussian formulae used for an approximate estimation of integrals. 2000 Mathematics Subject Classification: 33C45, 65D32.