We obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k−Toeplitz matrix A. The proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of Chebyshev polynomials of the second kind. We also compute the characteristic polynomial of A which enable us to s...

We construct an orthogonal wavelet basis for the interval using a linear combination of Legendre polynomials. The coefficients are taken as appropriate roots of Chebyshev polynomials of the second kind. The one-dimensional transform is applied to analytical data and appropriate definitions of a scalogram as well as local and global spectra are presented. The transform is then extended to the mu...

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...

formulae expressing explicitly the coefficients of an expansion of double jacobi polynomials which has been partially differentiated an arbitrary number of times with respect to its variables in terms of the coefficients of the original expansion are stated and proved. extension to expansion of triple jacobi polynomials is given. the results for the special cases of double and triple ultraspher...

The goal of this paper is to derive Hermite–Hadamard–Fejér-type inequalities for higher-order convex functions and a general three-point integral formula involving harmonic sequences polynomials w-harmonic functions. In special cases, estimates are derived various classical quadrature formulae such as the Gauss–Legendre Gauss–Chebyshev first second kind.

We show that the resultants with respect to x of certain linear forms in Chebyshev polynomials with argument x are again linear forms in Chebyshev polynomials. Their coefficients and arguments are certain rational functions of the coefficients of the original forms. We apply this to establish several related results involving resultants and discriminants of polynomials, including certain self-r...

A probabilistic interpretation of a modified Gegenbauer polynomial is furnished by its expression in terms of a combinatorial probability defined on a compound urn model. Also, a combinatorial interpretation of its coefficients is provided. In particular, probabilistic interpretations of a modified Chebyshev polynomial of the second kind and a modified Legendre polynomial together with combinat...

The purpose of this note is to extend in a simple and unified way some results on orthogonal polynomials with respect the weight function|Tm(x)|p1−x2,−1−1. Namely, provides an explicit representation recurrence coefficients for above function these terms known functions.

The paper presents the automated design of active RC and programmable filters. The approximating function is derived using Chebyshev orthogonal polynomials of the first kind. Optimization is performed using symbolic manipulation of expressions inputted into computer algebra system. The code is in the form of template notebook, and the user should specify the minimal number of numeric values, su...

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...