scholarly journals The Askey–Wilson polynomials and q-Sturm–Liouville problems

1996 ◽  
Vol 119 (1) ◽  
pp. 1-16 ◽  
Author(s):  
B. Malcolm Brown ◽  
W. Desmond Evans ◽  
Mourad E. H. Ismail
Keyword(s):  
Difference Operator ◽  
Divided Difference ◽  
Liouville Problem ◽  
Inner Product ◽  
Cauchy Principal ◽  
Wilson Operator ◽  
Principal Value ◽  
Sturm Liouville ◽  
Wilson Polynomials ◽  
Theoretic Description

AbstractWe find the adjoint of the Askey–Wilson divided difference operator with respect to the inner product on L2(–1, 1, (1– x2)½dx) defined as a Cauchy principal value and show that the Askey-Wilson polynomials are solutions of a q-Sturm–Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward way. We also provide an operator theoretic description of the Askey-Wilson operator.

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.

scholarly journals Spectral Analysis of -Sturm-Liouville Problem with the Spectral Parameter in the Boundary Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Aytekin Eryılmaz
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Spectral Parameter ◽  
Difference Operator ◽  
Boundary Value ◽  
Liouville Problem ◽  
Scattering Matrix ◽  
Eigenvalues And Eigenvectors ◽  
Sturm Liouville ◽  
Spectral Representations

This paper is concerned with -Sturm-Liouville boundary value problem in the Hilbert space with a spectral parameter in the boundary condition. We construct a self-adjoint dilation of the maximal dissipative -difference operator and its incoming and outcoming spectral representations, which make it possible to determine the scattering matrix of the dilation. We prove theorems on the completeness of the system of eigenvalues and eigenvectors of operator generated by boundary value problem.

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 the spectral 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 the q-Painlevé system.

scholarly journals Теория Титчмарша - Вейля сингулярного уравнения Хана - Штурма - Лиувилля

2021 ◽  
pp. 16-26
Author(s):  
B.P. Allahverdiev ◽  
H. Tuna
Keyword(s):  
Hilbert Space ◽  
Continuous Function ◽  
Difference Equation ◽  
Difference Operator ◽  
Liouville Problem ◽  
Limit Circle ◽  
Weyl Theory ◽  
Hahn Difference Operator ◽  
Sturm Liouville ◽  
Ordinary Derivative

In this work, we will consider the singular Hahn--Sturm--Liouville difference equation defined by $-q^{-1}D_{-\omega q^{-1},q^{-1}}D_{\omega ,q}y( x) +v(x) y( x) =\lambda y(x)$, $x\in (\omega _{0},\infty),$ where $\lambda$ is a complex parameter, $v$ is a real-valued continuous function at $\omega _{0}$ defined on $[\omega _{0},\infty)$. These type equations are obtained when the ordinary derivative in the classical Sturm--Liouville problem is replaced by the $\omega,q$-Hahn difference operator $D_{\omega,q}$. We develop the $\omega,q$-analogue of the classical Titchmarsh--Weyl theory for such equations. In other words, we study the existence of square-integrable solutions of the singular Hahn--Sturm--Liouville equation. Accordingly, first we define an appropriate Hilbert space in terms of Jackson--N\"{o}rlund integral and then we study families of regular Hahn--Sturm--Liouville problems on $[\omega_{0},q^{-n}]$, $n\in \mathbb{N}$. Then we define a family of circles that converge either to a point or a circle. Thus, we will define the limit-point, limit-circle cases in the Hahn calculus setting by using Titchmarsh's technique.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

Sign in / Sign up

Export Citation Format

Share Document