Classical orthogonal polynomials with weight function ((ax + b)2 + (cx + d)2)−pexp(qArctg((ax + b)/(cx + d))),x ∈ (−∞, ∞) and a generalization of T and F distributions

2004 ◽  
Vol 15 (2) ◽  
pp. 137-153 ◽  
Author(s):  
Mohammad Masjed Jamei
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Classical Orthogonal Polynomials

Jacobi series which converge to zero, with applications to a class of singular partial differential equations

1969 ◽  
Vol 65 (1) ◽  
pp. 101-106 ◽  
Author(s):  
Jet Wimp ◽  
David Colton
Keyword(s):  
Partial Differential Equations ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Hermite Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Applied Mathematician ◽  
In Series ◽  
Image Position ◽  
Series Of Functions ◽  
Sequence Of Functions

Expansions in series of functions are one of the most important tools of the applied mathematician, particularly expansions in series of the classical orthogonal polynomials, e.g. Laguerre, Jacobi and Hermite polynomials. In applied problems, the uniqueness of the particular expansion is usually intrinsic to the analysis, and often implicitly assumed. Indeed, in those cases where the functions in the series are orthogonal, uniqueness can often be proved by an argument that runs as follows. Let {φn(x)} (n = 0, 1, 2, …) be a sequence of functions orthogonal with respect to the weight function ρ(x) over the interval [0, 1], and suppose thatthe series being boundedly convergent for 0 ≤ x ≤ 1.

scholarly journals On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1250
Author(s):  
Abey S. Kelil ◽  
Appanah R. Appadu
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Distribution Of Zeros ◽  
Classical Orthogonal Polynomials ◽  
Ladder Operators ◽  
Recurrence Coefficients ◽  
Freud Weight ◽  
General Deformation ◽  
Linear Differential ◽  
Electrostatic Interpretation

Polynomials that are orthogonal with respect to a perturbation of the Freud weight function by some parameter, known to be modified Freudian orthogonal polynomials, are considered. In this contribution, we investigate certain properties of semi-classical modified Freud-type polynomials in which their corresponding semi-classical weight function is a more general deformation of the classical scaled sextic Freud weight |x|αexp(−cx6),c>0,α>−1. Certain characterizing properties of these polynomials such as moments, recurrence coefficients, holonomic equations that they satisfy, and certain non-linear differential-recurrence equations satisfied by the recurrence coefficients, using compatibility conditions for ladder operators for these orthogonal polynomials, are investigated. Differential-difference equations were also obtained via Shohat’s quasi-orthogonality approach and also second-order linear ODEs (with rational coefficients) satisfied by these polynomials. Modified Freudian polynomials can also be obtained via Chihara’s symmetrization process from the generalized Airy-type polynomials. The obtained linear differential equation plays an essential role in the electrostatic interpretation for the distribution of zeros of the corresponding Freudian polynomials.

scholarly journals On Koornwinder classical orthogonal polynomials in two variables

2012 ◽  
Vol 236 (15) ◽  
pp. 3817-3826 ◽  
Author(s):  
Lidia Fernández ◽  
Teresa E. Pérez ◽  
Miguel A. Piñar
Keyword(s):  
Orthogonal Polynomials ◽  
Classical Orthogonal Polynomials

Orthogonal polynomials: applications and computation

Acta Numerica ◽  
1996 ◽  
Vol 5 ◽  
pp. 45-119 ◽  
Author(s):  
Walter Gautschi
Keyword(s):  
Numerical Methods ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Rational Function ◽  
Recurrence Relations ◽  
Recurrence Coefficients ◽  
Basic Task ◽  
Sobolev Orthogonal Polynomials ◽  
Gauss Type ◽  
Three Term Recurrence Relation

We give examples of problem areas in interpolation, approximation, and quadrature, that call for orthogonal polynomials not of the classical kind. We then discuss numerical methods of computing the respective Gauss-type quadrature rules and orthogonal polynomials. The basic task is to compute the coefficients in the three-term recurrence relation for the orthogonal polynomials. This can be done by methods relying either on moment information or on discretization procedures. The effect on the recurrence coefficients of multiplying the weight function by a rational function is also discussed. Similar methods are applicable to computing Sobolev orthogonal polynomials, although their recurrence relations are more complicated. The paper concludes with a brief account of available software.

scholarly journals Orthogonal structure on a quadratic curve

2020 ◽  
Author(s):  
Sheehan Olver ◽  
Yuan Xu
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Extension Problem ◽  
Orthogonal Expansions ◽  
Schrödinger’S Equation ◽  
Fourier Extension ◽  
Quadratic Curve ◽  
Interpolation Of Functions ◽  
Fourier Orthogonal Expansions ◽  
Schrödinger's Equation

Abstract Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. We discuss applications to the Fourier extension problem, interpolation of functions with singularities or near singularities and the solution of Schrödinger’s equation with nondifferentiable or nearly nondifferentiable potentials.

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