Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials

1988 ◽  
pp. 261-278 ◽  
Author(s):  
Alphonse P. Maqnus
Keyword(s):  
Orthogonal Polynomials ◽  
Wilson Polynomials

Complex Weight Functions for Classical Orthogonal Polynomials

1991 ◽  
Vol 43 (6) ◽  
pp. 1294-1308 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
David R. Masson ◽  
Mizan Rahman
Keyword(s):  
Orthogonal Polynomials ◽  
Weight Functions ◽  
Classical Orthogonal Polynomials ◽  
The Real ◽  
Complex Functions ◽  
Wilson Polynomials ◽  
Distributional Weight

AbstractWe give complex weight functions with respect to which the Jacobi, Laguerre, little q-Jacobi and Askey-Wilson polynomials are orthogonal. The complex functions obtained are weight functions in a wider range of parameters than the real weight functions. They also provide an alternative to the recent distributional weight functions of Morton and Krall, and the more recent hyperfunction weight functions of Kim.

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).

On the Askey-Wilson and Rogers Polynomials

1988 ◽  
Vol 40 (5) ◽  
pp. 1025-1045 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
Dennis Stanton
Keyword(s):  
Orthogonal Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Wilson Polynomials ◽  
Image Position ◽  
Empty Product

The q-shifted factorial (a)n or (a; q)n isand an empty product is interpreted as 1. Recently, Askey and Wilson [6] introduced the polynomials1.1where1.2and1.3We shall refer to these polynomials as the Askey-Wilson polynomials or the orthogonal 4ϕ3 polynomials. They generalize the 6 — j symbols and are the most general classical orthogonal polynomials, [2].

A Product Formula and a Non-Negative Poisson Kernel for Racah-Wilson Polynomials

1980 ◽  
Vol 32 (6) ◽  
pp. 1501-1517 ◽  
Author(s):  
Mizan Rahman
Keyword(s):  
Orthogonal Polynomials ◽  
Series Representation ◽  
General System ◽  
Continuous Measure ◽  
Absolutely Continuous ◽  
Angular Momenta ◽  
Special Cases ◽  
Absolutely Continuous Measure ◽  
Wilson Polynomials ◽  
Special Case

Physicists have long been using Racah's [7] 6-j symbols as a representation for the addition coefficients of three angular momenta. Racah himself discovered a series representation of the 6-j symbol which can be expressed as a balanced 4F3 series of argument 1, that is, a generalized hypergeometric function such that the sum of the 3 denominator parameters exceeds that of the 4 numerator parameters by 1. What Racah does not seem to have realized or, perhaps, cared to investigate, is that his 4F3 functions, with variables and parameters suitably identified, form a system of orthogonal polynomials in a discrete variable. The orthogonality of 6-j symbols as an orthogonality of 4F3 polynomials was recognized much later by Biedenharn et al. [3] in some special cases. Recently J. Wilson [13, 14] introduced a very general system of orthogonal polynomials expressible as balanced 4F3 functions of argument 1 orthogonal with respect to an absolutely continuous measure and/or a discrete weight function. Wilson's polynomials contain Racah's 6-j symbols as a special case. These polynomials might rightfully be credited to Wilson alone, but justice might be better served if we call them Racah-Wilson polynomials.

Asymptotics of the Wilson polynomials

2019 ◽  
Vol 18 (02) ◽  
pp. 237-270 ◽  
Author(s):  
Yu-Tian Li ◽  
Xiang-Sheng Wang ◽  
Roderick Wong
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Orthogonal Polynomials ◽  
Difference Equations ◽  
Asymptotic Expansions ◽  
Transition Point ◽  
Linear Difference Equations ◽  
Variable Formula ◽  
Free Zone ◽  
Wilson Polynomials

In this paper, we study the asymptotic behavior of the Wilson polynomials [Formula: see text] as their degree tends to infinity. These polynomials lie on the top level of the Askey scheme of hypergeometric orthogonal polynomials. Infinite asymptotic expansions are derived for these polynomials in various cases, for instance, (i) when the variable [Formula: see text] is fixed and (ii) when the variable is rescaled as [Formula: see text] with [Formula: see text]. Case (ii) has two subcases, namely, (a) zero-free zone ([Formula: see text]) and (b) oscillatory region [Formula: see text]. Corresponding results are also obtained in these cases (iii) when [Formula: see text] lies in a neighborhood of the transition point [Formula: see text], and (iv) when [Formula: see text] is in the neighborhood of the transition point [Formula: see text]. The expansions in the last two cases hold uniformly in [Formula: see text]. Case (iv) is also the only unsettled case in a sequence of works on the asymptotic analysis of linear difference equations.

scholarly journals Multiple Askey–Wilson polynomials and related basic hypergeometric multiple orthogonal polynomials

2020 ◽  
Vol 373 (12) ◽  
pp. 8289-8312
Author(s):  
Jean Paul Nuwacu ◽  
Walter Van Assche
Keyword(s):  
Orthogonal Polynomials ◽  
Multiple Orthogonal Polynomials ◽  
Wilson Polynomials

Limit Transitions for BC Type Multivariable Orthogonal Polynomials

1997 ◽  
Vol 49 (2) ◽  
pp. 374-405 ◽  
Author(s):  
Jasper V. Stokman ◽  
Tom H. Koornwinder
Keyword(s):  
Orthogonal Polynomials ◽  
Root System ◽  
Jacobi Polynomials ◽  
Parameter Family ◽  
Difference Operators ◽  
Wilson Polynomials ◽  
The One

AbstractLimit transitions will be derived between the five parameter family of Askey-Wilson polynomials, the four parameter family of big q-Jacobi polynomials and the three parameter family of little q-Jacobi polynomials in n variables associated with root system BC. These limit transitions generalize the known hierarchy structure between these families in the one variable case. Furthermore it will be proved that these three families are q-analogues of the three parameter family of BC type Jacobi polynomials in n variables. The limit transitions will be derived by taking limits of q-difference operators which have these polynomials as eigenfunctions.

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