Two-dimensional stability and orthogonal polynomials on the hypercircle

‎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 this note, we propose a new class of orthogonal polynomials (named Bachok-Hasham polynomials of the first and second kind for order k, denote it as Z(i,n)k(x), i={ 1,2 }, which is extension of the Chebyshev polynomials of the first and second kind respectively. It is found that Bachok--Hasham polynomials of first and second kind Z(i,n)k(x) are orthogonal with respect to weights w(1,k)(x)=xk−11−x2k, w(2,k)(x)=xk−11−x2k on the interval [-1,1], where k is positive odd integers. Spectral properties Bachok--Hasham polynomials of the first and second kind Z(i,n)k(x),i={ 1,2 } are proved. These properties are used to solve a special class of singular integral equations. Finally, numerical examples and comparison results with other methods are provided to illustrate the effectiveness and accuracy of the proposed method.

Received by the editors June 23, 1952. 1 H. L. Krall and Orrin Frink, A new class of orthogonal polynomials; the Bessel polynomials, Trans. Amer. Math. Soc. vol. 65 (1949) pp. 100-115. 2 See J. L. Burchnall, The Bessel polynomials, Canadian Journal of Mathematics vol. 3 (1951) pp. 62-68, and E. D. Rainville, Generating functions for Bessel and related polynomials, Canadian Journal of Mathematics vol. 5 (1953) pp. 104-106. 3 Each 2Fo in (3) diverges except for nonpositive integer parameters or for zero argument. Equation (3) means that the coefficient of each power of t on the left equals the coefficient of the corresponding power on the right. Fortunately convergence is not in general necessary in using generating functions.

A New Class of Orthogonal Polynomials: The Bessel Polynomials

In this paper, we determine all pairs of orthogonal polynomial sequences $\{ p_n (x)\} $ and $\{ q_n (x)\} $, such that their convolution, \[ Q_n (x,y) = \sum_{k = 0}^n {p_k (x)q_{n - k} (y)} ,\quad n \geqq 0.\] defines $\{ Q_n (x,y)\} $ as an orthogonal polynomial sequence in x for all y. All such triples are determined explicitly in terms of their three-term recurrence formulas. Generating functions and “explicit” representation formulas are obtained. The resulting sequences are found to consist of a class of orthogonal polynomials characterized by J. Meixner (which class includes the Laguerre and Hermite polynomials) together with a new class of orthogonal polynomials which includes the orthogonal q-polynomials of Al-Salam and Carlitz. Explicit orthogonality relations are found for one new special case of this latter class.

Weak classical orthogonal polynomials in two variables

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 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.

The direct integration of second order initial and boundary value problems is considered in this paper. We employ a new class of orthogonal polynomials constructed as basis function to develop a two-step hybrid block method (2SHBM) adopting collocation technique. The recursive formula of the class of polynomials have been constructed, and then we give analysis of the basic properties of 2SHBM as findings show that the method is accurate and convergent. The boundary locus of the proposed 2SHBM shows that the new scheme is A-stable.

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 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.

In this paper, linear multi-step hybrid block methods with three-, four- and five-step numbers are developed for approximating directly the solution of second order Initial and Boundary Value Problems (IBVPs). Multiple finite difference formulas are derived and combined in a block formulation to form a numerical integrator that provides direct solution to second order IBVPs over sub-intervals. A new class of orthogonal polynomials constructed as basis function to develop the hybrid block methods adopting collocation technique with a non-negative weight function. The scheme is applied as simultaneous integrator to second order initial value and boundary value problems of ODEs. The properties and convergence of the proposed method are discussed. The derived schemes were used to solve some problems and the numerical result shows the effectiveness, accuracy and superiority of the method over the existing methods found in the literature.

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

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