Characterizations of Continuous and Discrete q-Ultraspherical Polynomials

2011 ◽  
Vol 63 (1) ◽  
pp. 181-199 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
Josef Obermaier
Keyword(s):  
Special Form ◽  
Jacobi Polynomials ◽  
Difference Operator ◽  
Divided Difference ◽  
Ultraspherical Polynomials ◽  
Connection Coefficients ◽  
Similar Characterization

Abstract We characterize the continuous q-ultraspherical polynomials in terms of the special form of the coefficients in the expansion DqPn(x) in the basis {Pn(x)}, Dq being the Askey-Wilson divided difference operator. The polynomials are assumed to be symmetric, and the connection coefficients are multiples of the reciprocal of the square of the L2 norm of the polynomials. A similar characterization is given for the discrete q-ultraspherical polynomials. A new proof of the evaluation of the connection coefficients for big q-Jacobi polynomials is given.

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 Stability Anomalies of Some Jacobian-Free Iterative Methods of High Order of Convergence

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 51
Author(s):  
Alicia Cordero ◽  
Javier G. Maimó ◽  
Juan R. Torregrosa ◽  
María P. Vassileva
Keyword(s):  
Difference Operator ◽  
Divided Difference ◽  
Polynomial System ◽  
Nonlinear Problems ◽  
Coefficient Matrix ◽  
Test Problems ◽  
Stability Performance ◽  
Need To Evaluate ◽  
The Stability ◽  
Theoretical Results

In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The stability performance of the classes is analyzed on a quadratic polynomial system, and it is shown that for many values of the parameter, only convergence to the roots of the problem exists. Finally, we check the performance of these methods on some test problems to confirm the theoretical results.

A Relation Between Ultraspherical and Jacobi Polynomial Sets

1953 ◽  
Vol 5 ◽  
pp. 301-305 ◽  
Author(s):  
Fred Brafman
Keyword(s):  
Jacobi Polynomial ◽  
Legendre Polynomials ◽  
Jacobi Polynomials ◽  
Ultraspherical Polynomials ◽  
Image Position ◽  
Special Case

The Jacobi polynomials may be defined bywhere (a)n = a (a + 1) … (a + n — 1). Putting β = α gives the ultraspherical polynomials which have as a special case the Legendre polynomials .

scholarly journals A New Class of Iterative Processes for Solving Nonlinear Systems by Using One Divided Differences Operator

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 776 ◽  
Author(s):  
Alicia Cordero ◽  
Cristina Jordán ◽  
Esther Sanabria ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Boundary Problem ◽  
Fourth Order ◽  
Difference Operator ◽  
Divided Difference ◽  
Order Convergence ◽  
New Class ◽  
Numerical Tests ◽  
New Family ◽  
Theoretical Results

In this manuscript, a new family of Jacobian-free iterative methods for solving nonlinear systems is presented. The fourth-order convergence for all the elements of the class is established, proving, in addition, that one element of this family has order five. The proposed methods have four steps and, in all of them, the same divided difference operator appears. Numerical problems, including systems of academic interest and the system resulting from the discretization of the boundary problem described by Fisher’s equation, are shown to compare the performance of the proposed schemes with other known ones. The numerical tests are in concordance with the theoretical results.

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 Semilocal Convergence of the Extension of Chun’s Method

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 161
Author(s):  
Alicia Cordero ◽  
Javier G. Maimó ◽  
Eulalia Martínez ◽  
Juan R. Torregrosa ◽  
María P. Vassileva
Keyword(s):  
Iterative Method ◽  
Existence And Uniqueness ◽  
Recurrence Relations ◽  
Nonlinear Integral Equation ◽  
Difference Operator ◽  
Divided Difference ◽  
Semilocal Convergence ◽  
Starting Point ◽  
Domain Of Existence ◽  
Theoretical Results

In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun’s iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Fréchet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.

scholarly journals Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy

2015 ◽  
Vol 219 ◽  
pp. 127-234 ◽  
Author(s):  
N. S. Witte
Keyword(s):  
Orthogonal Polynomial ◽  
Difference Operator ◽  
Divided Difference ◽  
Polynomial System ◽  
Homogeneous Equation ◽  
Polynomial Systems ◽  
Classical Solutions ◽  
First Order ◽  
Wilson Operator ◽  
Spectral Equation

AbstractA 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed thespectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to theq-Painlevé system.

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