scholarly journals On the Complex Zeros of Some Families of Orthogonal Polynomials

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Eugenia N. Petropoulou
Keyword(s):  
Asymptotic Behavior ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Laguerre Polynomials ◽  
The Real ◽  
Ultraspherical Polynomials ◽  
Complex Zeros

The complex zeros of the orthogonal Laguerre polynomials for , ultraspherical polynomials for , Jacobi polynomials for , , , orthonormal Al-Salam-Carlitz II polynomials for , , and -Laguerre polynomials for , are studied. Several inequalities regarding the real and imaginary properties of these zeros are given, which help locating their position. Moreover, a few limit relations regarding the asymptotic behavior of these zeros are proved. The method used is a functional analytic one. The obtained results complement and improve previously known results.

Classical Orthogonal Polynomials as Moments

1997 ◽  
Vol 49 (3) ◽  
pp. 520-542 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
Dennis Stanton
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Laguerre Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Ultraspherical Polynomials

AbstractWe show that the Meixner, Pollaczek, Meixner-Pollaczek, the continuous q-ultraspherical polynomials and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures.We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials.

scholarly journals Connection Problem for Sums of Finite Products of Legendre and Laguerre Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 317 ◽  
Author(s):  
Taekyun Kim ◽  
Kyung-Won Hwang ◽  
Dae Kim ◽  
Dmitry Dolgy
Keyword(s):  
Orthogonal Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Laguerre Polynomials ◽  
Linear Combinations ◽  
Connection Problem ◽  
Finite Products

The purpose of this paper is to represent sums of finite products of Legendre and Laguerre polynomials in terms of several orthogonal polynomials. Indeed, by explicit computations we express each of them as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials, some of which involve terminating hypergeometric functions 1 F 1 and 2 F 1 .

scholarly journals New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 74
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Afnan Ali
Keyword(s):  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Algebraic Computation ◽  
Ultraspherical Polynomials ◽  
Current Article ◽  
Linearization Coefficients

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

scholarly journals Sylvester equations for Laguerre–Hahn orthogonal polynomials on the real line

2013 ◽  
Vol 219 (17) ◽  
pp. 9118-9131 ◽  
Author(s):  
A. Branquinho ◽  
A. Paiva ◽  
M.N. Rebocho
Keyword(s):  
Orthogonal Polynomials ◽  
Real Line ◽  
The Real

scholarly journals LOWERING AND RAISING OPERATORS FOR THE FREE MEIXNER CLASS OF ORTHOGONAL POLYNOMIALS

2009 ◽  
Vol 12 (03) ◽  
pp. 387-399 ◽  
Author(s):  
EUGENE LYTVYNOV ◽  
IRINA RODIONOVA
Keyword(s):  
Orthogonal Polynomials ◽  
Real Line ◽  
Meixner Polynomials ◽  
The Real ◽  
Raising Operators ◽  
Meixner Class

We compare some properties of the lowering and raising operators for the classical and free classes of Meixner polynomials on the real line.

scholarly journals Exact solvability of two new 3D and 1D non-relativistic potentials within the TRA framework

2018 ◽  
Vol 33 (32) ◽  
pp. 1850187 ◽  
Author(s):  
I. A. Assi ◽  
H. Bahlouli ◽  
A. Hamdan
Keyword(s):  
Wave Function ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Basis Functions ◽  
Exact Solvability ◽  
Analytical Properties ◽  
Expansion Coefficients ◽  
Potential Parameters ◽  
Square Integrable Basis

This work aims at introducing two new solvable 1D and 3D confined potentials and present their solutions using the Tridiagonal Representation Approach (TRA). The wave function is written as a series in terms of square integrable basis functions which are expressed in terms of Jacobi polynomials. The expansion coefficients are then written in terms of new orthogonal polynomials that were introduced recently by Alhaidari, the analytical properties of which are yet to be derived. Moreover, we have computed the numerical eigenenergies for both potentials by considering specific choices of the potential parameters.

Orthogonal Polynomials with Symmetry of Order Three

1984 ◽  
Vol 36 (4) ◽  
pp. 685-717 ◽  
Author(s):  
Charles F. Dunkl
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Sphere ◽  
Jacobi Polynomials ◽  
Orthogonal Basis ◽  
Lower Degree ◽  
Surface Measure ◽  
Rotation Invariant ◽  
General Measure ◽  
Invariant Surface ◽  
Image Position

The measure (x1x2x3)2adm(x) on the unit sphere in R3 is invariant under sign-changes and permutations of the coordinates; here dm denotes the rotation-invariant surface measure. The more general measurecorresponds to the measureon the triangle(where ). Appell ([1] Chap. VI) constructed a basis of polynomials of degree n in v1, v2 orthogonal to all polynomials of lower degree, and a biorthogonal set for the case γ = 0. Later Fackerell and Littler [6] found a biorthogonal set for Appell's polynomials for γ ≠ 0. Meanwhile Pronol [10] had constructed an orthogonal basis in terms of Jacobi polynomials.

A one dimensional random space-filling problem

1968 ◽  
Vol 5 (02) ◽  
pp. 427-435 ◽  
Author(s):  
John P. Mullooly
Keyword(s):  
Asymptotic Behavior ◽  
Finite Number ◽  
Real Line ◽  
Random Variables ◽  
Random Interval ◽  
Space Filling ◽  
One Dimensional ◽  
The Real ◽  
Fixed Constant ◽  
Filling Problem

Consider an interval of the real line (0, x), x > 0; and place in it a random subinterval S(x) defined by the random variables Xx and Yx , the position of the center of S(x) and the length of S(x). The set (0, x)– S(x) consists of two intervals of length δ and η. Let a > 0 be a fixed constant. If δ ≦ a, then a random interval S(δ) defined by Xδ, Yδ is placed in the interval of length δ. If δ < a, the placement of the second interval is not made. The same is done for the interval of length η. Continue to place non-intersecting random subintervals in (0, x), and require that the lengths of all the random subintervals be ≦ a. The process terminates after a finite number of steps when all the segments of (0, x) uncovered by random subintervals are of length < a. At this stage, we say that (0, x) is saturated. Define N(a, x) as the number of random subintervals that have been placed when the process terminates. We are interested in the asymptotic behavior of the moments of N(a, x), for large x.

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