Orthogonal polynomials derived from the tridiagonal representation approach

Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case

The zeros and asymptotic limits of two new classes of orthogonal polynomials, which are derived by applying two orthogonalization procedures due to Lowdin to a set of monomials, are calculated. It is established that they possess all the properties ofthe zeros of a polynomial. Their asymptotic limits are found. A Unified view of all the Lowdin orthogonal polynomials together with the standard classical orthogonal polynomials are presented in a unique graph.

We consider the eigenvalue problem associated with the Dunkl-type differential operator (in which the reflection operator R is involved)in the context of supersymmetric quantum mechanical models. By solving this eigenvalue problem with the help of known exactly solvable potentials, we construct several classes of functions satisfying certain orthogonality relations. We call them the Dunkl-supersymmetric (Dunkl-SUSY) orthogonal functions. These functions can be expressed in terms of the classical orthogonal polynomials. The key feature of these functions is that they appear by pairs, i.e. Qn(x) and Qn(−x) are both the eigenfunctions of . A general formulation of the Dunkl-SUSY orthogonal polynomials is also presented.

We derive and factorize the fourth-order q-difference equations satisfied by orthogonal polynomials obtained from some perturbations of the recurrence coefficients of q-classical orthogonal polynomials. These perturbations include the rth associated, the anti-associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified q-classical orthogonal polynomials. Moreover we find a basis of four linearly independent solutions of these fourth-order q-difference equations and express the modified families in terms of the starting ones.

This paper is concerned with a possible extension of a well-known stabilization technique for one-variable recursive digital filters to the two-dimensional case, as recently conjectured. It is shown that this problem is equivalent to considering a new class of orthogonal polynomials, the two-variable orthogonal polynomials on the hypercircle, the properties of which are investigated. As a result, the zeros of these polynomials are proved not to lie necessarily in an appropriate region compatible with the proposed conjecture, which therefore turns out to be in error.

This paper continues the analysis of a class of orthogonal polynomials in two variables on a region bounded by two straight lines and a parabola touching these lines, which was introduced by the first author. An explicit series expansion for these polynomials is obtained, which generalizes Constantine’s expansion of hypergeometric functions of $(2 \times 2)$ matrix argument in terms of James’ zonal polynomials. In two special cases the orthogonal polynomials turn out to be Appell’s hypergeometric $F_4 $-functions and certain hypergeometric functions in two variables of order three, respectively.

In this paper, we will present a strong (or pathwise) approximation of\nstandard Brownian motion by a class of orthogonal polynomials. The coefficients\nthat are obtained from the expansion of Brownian motion in this polynomial\nbasis are independent Gaussian random variables. Therefore it is practical\n(requires $N$ independent Gaussian coefficients) to generate an approximate\nsample path of Brownian motion that respects integration of polynomials with\ndegree less than $N$. Moreover, since these orthogonal polynomials appear\nnaturally as eigenfunctions of an integral operator defined by the Brownian\nbridge covariance function, the proposed approximation is optimal in a certain\nweighted $L^{2}(\\mathbb{P})$ sense. In addition, discretizing Brownian paths as\npiecewise parabolas gives a locally higher order numerical method for\nstochastic differential equations (SDEs) when compared to the standard\npiecewise linear approach. We shall demonstrate these ideas by simulating\nInhomogeneous Geometric Brownian Motion (IGBM). This numerical example will\nalso illustrate the deficiencies of the piecewise parabola approximation when\ncompared to a new version of the asymptotically efficient log-ODE (or\nCastell-Gaines) method.\n

‎In this article‎, ‎an efficient numerical method based on a new class of orthogonal polynomials‎, ‎namely Chelyshkov polynomials‎, ‎has been presented to approximate solution of time-fractional telegraph (TFT) equations‎. ‎The fractional operational matrix of the Chelyshkov polynomials along with the typical collocation method is used to reduces TFT equations to a system of algebraic equations‎. ‎The error analysis of the proposed collocation method is also investigated‎. ‎A comparison with other published results confirms that the presented Chelyshkov collocation approach is efficient and accurate for solving TFT equations‎. ‎Illustrative examples are included to demonstrate the efficiency of the Chelyshkov method‎.

In applications of mathematics involving either the Laplace or the Helmholtz equation in spherical coordinates the associated Legendre equation occurs. Its solutions are called associated Legendre functions. They have some relations to classical Legendre orthogonal polynomials and classical Jacobi orthogonal polynomials. The classical Legendre polynomials and the associated Legendre functions have been used by some authors at the approximation of length of curves and cylindrical surfaces and the approximation of Earth shape in geosciences and engineering computations. In the present paper we investigate some relations between these functions pointing out possible applications in geosciences.

Orthogonal polynomials arising from the wreath products of a dihedral group with a symmetric group

Orthogonal polynomials suggested by a queueing model

Weak classical orthogonal polynomials in two variables

Fourth-order difference equation for the associated classical discrete orthogonal polynomials

Connection and linearization coefficients of the Askey–Wilson polynomials

Quadrature formulas for Bessel polynomials