Fourier coefficients for Laguerre–Sobolev type orthogonal polynomials

Purpose In this paper, the authors take the first step in the study of constructive methods by using Sobolev polynomials.Design/methodology/approach To do that, the authors use the connection formulas between Sobolev polynomials and classical Laguerre polynomials, as well as the well-known Fourier coefficients for these latter.Findings Then, the authors compute explicit formulas for the Fourier coefficients of some families of Laguerre–Sobolev type orthogonal polynomials over a finite interval. The authors also describe an oscillatory region in each case as a reasonable choice for approximation purposes.Originality/value In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product. As far as the authors know, this particular problem has not been addressed in the existing literature.

Filter integrals for orthogonal polynomials

Motivated by an expression by Persson and Strang on an integral involving Legendre polynomials, stating that the square of $P_{2n+1}(x)/x$ integrated over $[-1,1]$ is always $2$, we present analog results for Hermite, Chebyshev, Laguerre and Gegenbauer polynomials as well as the original Legendre polynomial with even index.

ESTIMATING WITH A GIVEN ACCURACY OF THE COEFFICIENTS AT NONLINEAR TERMS OF UNIVARIATE POLYNOMIAL REGRESSION USING A SMALL NUMBER OF TESTS IN AN ARBITRARY LIMITED ACTIVE EXPERIMENT

We substantiate the structure of the efficient numerical axis segment an active experiment on which allows finding estimates of the coefficients fornonlinear terms of univariate polynomial regression with high accuracy using normalized orthogonal Forsyth polynomials with a sufficiently smallnumber of experiments. For the case when an active experiment can be executed on a numerical axis segment that does not satisfy these conditions, wesubstantiate the possibility of conducting a virtual active experiment on an efficient interval of the numerical axis. According to the results of the experiment, we find estimates for nonlinear terms of the univariate polynomial regression under research as a solution of a linear equalities system withan upper non-degenerate triangular matrix of constraints. Thus, to solve the problem of estimating the coefficients for nonlinear terms of univariatepolynomial regression, it is necessary to choose an efficient interval of the numerical axis, set the minimum required number of values of the scalarvariable which belong to this segment and guarantee a given value of the variance of estimates for nonlinear terms of univariate polynomial regressionusing normalized orthogonal polynomials of Forsythe. Next, it is necessary to find with sufficient accuracy all the coefficients of the normalized orthogonal polynomials of Forsythe for the given values of the scalar variable. The resulting set of normalized orthogonal polynomials of Forsythe allows us to estimate with a given accuracy the coefficients of nonlinear terms of univariate polynomial regression in an arbitrary limited active experiment: the range of the scalar variable values can be an arbitrary segment of the numerical axis. We propose to find an estimate of the constant and ofthe coefficient at the linear term of univariate polynomial regression by solving the linear univariate regression problem using ordinary least squaresmethod in active experiment conditions. Author and his students shown in previous publications that the estimation of the coefficients for nonlinearterms of multivariate polynomial regression is reduced to the sequential construction of univariate regressions and the solution of the correspondingsystems of linear equalities. Thus, the results of the paper qualitatively increase the efficiency of finding estimates of the coefficients for nonlinearterms of multivariate polynomial regression given by a redundant representation.

New Results of the Fifth-Kind Orthogonal Chebyshev Polynomials

The principal objective of this article is to develop new formulas of the so-called Chebyshev polynomials of the fifth-kind. Some fundamental properties and relations concerned with these polynomials are proposed. New moments formulas of these polynomials are obtained. Linearization formulas for these polynomials are derived using the moments formulas. Connection problems between the fifth-kind Chebyshev polynomials and some other orthogonal polynomials are explicitly solved. The linking coefficients are given in forms involving certain generalized hypergeometric functions. As special cases, the connection formulas between Chebyshev polynomials of the fifth-kind and the well-known four kinds of Chebyshev polynomials are shown. The linking coefficients are all free of hypergeometric functions.

Orthogonal polynomials and generalized Gauss-Rys quadrature formulae

Orthogonal polynomials and the corresponding quadrature formulas of Gaussian type concerni λ ng > the 1 e / v 2 en wei x gh > t f 0 unction ω(t; x) = exp λ (−= xt 1 2) / ( 2 1 − t2)−1/2 on (−1, 1), with parameters − and , are considered. For these quadrature rules reduce to the socalled Gauss-Rys quadrature formulas, which were investigated earlier by several authors, e.g., Dupuis at al 1976 and 1983; Sagar 1992; Schwenke 2014; Shizgal 2015; King 2016; Milovanovic ´ 2018, etc. In this generalized case, the method of modified moments is used, as well as a transformation of quadratures on (−1, 1) with N nodes to ones on (0, 1) with only (N + 1)/2 nodes. Such an approach provides a stable and very efficient numerical construction.

Supersymmetry of 𝒫𝒯-symmetric tridiagonal Hamiltonians

We extend the study of supersymmetric tridiagonal Hamiltonians to the case of non-Hermitian Hamiltonians with real or complex conjugate eigenvalues. We find the relation between matrix elements of the non-Hermitian Hamiltonian [Formula: see text] and its supersymmetric partner [Formula: see text] in a given basis. Moreover, the orthogonal polynomials in the eigenstate expansion problem attached to [Formula: see text] can be recovered from those polynomials arising from the same problem for [Formula: see text] with the help of kernel polynomials. Besides its generality, the developed formalism in this work is a natural home for using the numerically powerful Gauss quadrature techniques in probing the nature of some physical quantities such as the energy spectrum of [Formula: see text]-symmetric complex potentials. Finally, we solve the shifted [Formula: see text]-symmetric Morse oscillator exactly in the tridiagonal representation.