Systems of orthogonal polynomials explicitly represented by the Jacobi polynomials

1987 ◽  
Vol 42 (5) ◽  
pp. 858-863 ◽  
Author(s):  
V. M. Badkov
Keyword(s):  
Orthogonal Polynomials ◽  
Jacobi Polynomials

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.

scholarly journals Zeros of the Exceptional Laguerre and Jacobi Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Choon-Lin Ho ◽  
Ryu Sasaki
Keyword(s):  
Numerical Analysis ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Mathematical Physics ◽  
Positive Definite ◽  
Classical Orthogonal Polynomials ◽  
Complete Sets

An interesting discovery in the last two years in the field of mathematical physics has been the exceptional Xℓ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree ℓ=1,2,…, and yet they form complete sets with respect to some positive-definite measure. In this paper, we study one important aspect of these new polynomials, namely, the behaviors of their zeros as some parameters of the Hamiltonians change. Most results are of heuristic character derived by numerical analysis.

Time-and-band limiting for matrix orthogonal polynomials of Jacobi type

2017 ◽  
Vol 06 (04) ◽  
pp. 1740001 ◽  
Author(s):  
M. Castro ◽  
F. A. Grünbaum
Keyword(s):  
Signal Processing ◽  
Differential Operator ◽  
Orthogonal Polynomials ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Jacobi Polynomials ◽  
Matrix Theory ◽  
Matrix Orthogonal Polynomials

We extend to a situation involving matrix-valued orthogonal polynomials a scalar result that plays an important role in Random Matrix Theory and a few other areas of mathe-matics and signal processing. We consider a case of matrix-valued Jacobi polynomials which arises from the study of representations of [Formula: see text], a group that plays an important role in Random Matrix Theory. We show that in this case an algebraic miracle, namely the existence of a differential operator that commutes with a naturally arising integral one, extends to this matrix-valued situation.

scholarly journals SPECTRAL PROPERTIES OF OPERATORS USING TRIDIAGONALIZATION

2012 ◽  
Vol 10 (03) ◽  
pp. 327-343 ◽  
Author(s):  
MOURAD E. H. ISMAIL ◽  
ERIK KOELINK
Keyword(s):  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Difference Operator ◽  
General Scheme ◽  
Second Order ◽  
Order Differential Operator ◽  
Difference Operators ◽  
Jacobi Function ◽  
Wilson Polynomials ◽  
Tridiagonal Form

A general scheme for tridiagonalizing differential, difference or q-difference operators using orthogonal polynomials is described. From the tridiagonal form the spectral decomposition can be described in terms of the orthogonality measure of generally different orthogonal polynomials. Three examples are worked out: (1) related to Jacobi and Wilson polynomials for a second order differential operator, (2) related to little q-Jacobi polynomials and Askey–Wilson polynomials for a bounded second order q-difference operator, (3) related to little q-Jacobi polynomials for an unbounded second order q-difference operator. In case (1) a link with the Jacobi function transform is established, for which we give a q-analogue using example (2).

scholarly journals Generalized hypergroups and orthogonal polynomials

2007 ◽  
Vol 82 (3) ◽  
pp. 369-394 ◽  
Author(s):  
Rupert Lasser ◽  
Josef Obermaier ◽  
Holger Rauhut
Keyword(s):  
Fourier Analysis ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
The Relationship ◽  
Discrete Hypergroups

AbstractThe concept of semi-bounded generalized hypergroups (SBG hypergroups) is developed. These hypergroups are more special than generalized hypergroups introduced by Obata and Wildberger and more general than discrete hypergroups or even discrete signed hypergroups. The convolution of measures and functions is studied. In the case of commutativity we define the dual objects and prove some basic theorems of Fourier analysis. Furthermore, we investigate the relationship between orthogonal polynomials and generalized hypergroups. We discuss the Jacobi polynomials as an example.

scholarly journals ON CONNECTING WEYL-ORBIT FUNCTIONS TO JACOBI POLYNOMIALS AND MULTIVARIATE (ANTI)SYMMETRIC TRIGONOMETRIC FUNCTIONS

2016 ◽  
Vol 56 (4) ◽  
pp. 283-290 ◽  
Author(s):  
Jiri Hrivnak ◽  
Lenka Motlochova
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Jacobi Polynomials ◽  
Special Functions ◽  
The Other ◽  
Trigonometric Functions ◽  
The One ◽  
Sine Functions

<p>The aim of this paper is to make an explicit link between the Weyl-orbit functions and the corresponding polynomials, on the one hand, and to several other families of special functions and orthogonal polynomials on the other. The cornerstone is the connection that is made between the one-variable orbit functions of <em>A<sub>1</sub></em> and the four kinds of Chebyshev polynomials. It is shown that there exists a similar connection for the two-variable orbit functions of <em>A<sub>2</sub></em> and a specific version of two variable Jacobi polynomials. The connection with recently studied <em>G<sub>2</sub></em>-polynomials is established. Formulas for connection between the four types of orbit functions of <em>B<sub>n</sub></em> or <em>C<sub>n</sub></em> and the (anti)symmetric multivariate cosine and sine functions are explicitly derived.</p>

A Non-Negative Representation of the Linearization Coefficients of the Product of Jacobi Polynomials

1981 ◽  
Vol 33 (4) ◽  
pp. 915-928 ◽  
Author(s):  
Mizan Rahman
Keyword(s):  
Orthogonal Polynomials ◽  
Hypergeometric Function ◽  
Jacobi Polynomials ◽  
Defining Relation ◽  
Linearization Problem ◽  
Image Position ◽  
Linearization Coefficients ◽  
Negative Representation

The problem of linearizing products of orthogonal polynomials, in general, and of ultraspherical and Jacobi polynomials, in particular, has been studied by several authors in recent years [1, 2, 9, 10, 13-16]. Standard defining relation [7, 18] for the Jacobi polynomials is given in terms of an ordinary hypergeometric function:with Re α > –1, Re β > –1, –1 ≦ x ≦ 1. However, for linearization problems the polynomials Rn(α,β)(x), normalized to unity at x = 1, are more convenient to use:(1.1)Roughly speaking, the linearization problem consists of finding the coefficients g(k, m, n; α,β) in the expansion(1.2)

scholarly journals Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space

2004 ◽  
Vol 2004 (52) ◽  
pp. 2761-2772 ◽  
Author(s):  
Fred Brackx ◽  
Nele De Schepper ◽  
Frank Sommen
Keyword(s):  
Euclidean Space ◽  
Orthogonal Polynomials ◽  
Clifford Algebra ◽  
Unit Ball ◽  
Real Axis ◽  
Jacobi Polynomials ◽  
Orthogonality Relation ◽  
Weight Functions ◽  
Open Unit ◽  
Open Unit Ball

A new method for constructing Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space is presented. In earlier research, we only dealt with scalar-valued weight functions. Now the class of weight functions involved is enlarged to encompass Clifford algebra-valued functions. The method consists in transforming the orthogonality relation on the open unit ball into an orthogonality relation on the real axis by means of the so-called Clifford-Heaviside functions. Consequently, appropriate orthogonal polynomials on the real axis give rise to Clifford algebra-valued orthogonal polynomials in the unit ball. Three specific examples of such orthogonal polynomials in the unit ball are discussed, namely, the generalized Clifford-Jacobi polynomials, the generalized Clifford-Gegenbauer polynomials, and the shifted Clifford-Jacobi polynomials.

scholarly journals Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 617 ◽  
Author(s):  
Dmitry Dolgy ◽  
Dae Kim ◽  
Taekyun Kim ◽  
Jongkyum Kwon
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Linear Combinations ◽  
The Third ◽  
Connection Problem ◽  
Finite Products

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions F 0 2 , F 1 2 , and F 2 3 .

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