Orthogonal polynomial solutions of linear ordinary differential equations

Everitt, W. N.; Kwon, Kil Hyun; Littlejohn, Lance L.; Wellman, R.

doi:10.1016/s0377-0427(00)00636-1

Cited by 62 publications

(51 citation statements)

References 63 publications

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“…We refer the reader to [14] and [15] for the precise formulation of the results in the context of the Laguerre and the Jacobi polynomials respectively. For a sample of other works related to Krall's problem or the role of the Darboux transformation in the theory of orthogonal polynomials, the reader can consult [9], [17], [19], [24], [27], [30], [37] and [44].…”

Section: Introductionmentioning

confidence: 99%

Askey-Wilson type functions with bound states

Haine

Илиев

2006

Ramanujan J

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Abstract. The two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22], are slightly modified so as to make it transparent that these functions satisfy a beautiful symmetry property. It essentially means that the geometric and the spectral parameters are interchangeable in these functions. We call the resulting functions the Askey-Wilson functions. Then, we show that by adding bound states (with arbitrary weights) at specific points outside of the continuous spectrum of some instances of the Askey-Wilson difference operator, we can generate functions that satisfy a doubly infinite three-term recursion relation and are also eigenfunctions of q-difference operators of arbitrary orders. Our result provides a discrete analogue of the solutions of the purely differential version of the bispectral problem that were discovered in the pioneering work [8] of Duistermaat and Grünbaum.

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Section: Introductionmentioning

confidence: 99%

Askey-Wilson type functions with bound states

Haine

Илиев

2006

Ramanujan J

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“…In [4] it is conjectured that the leading coefficient a N for any Bochner-Krall system is a power of a polynomial of degree at most 2. Our main result is an affirmative answer to this conjecture for Bochner-Krall systems of Nevai type.…”

Section: Discussionmentioning

confidence: 99%

“…A complete classification is only known for Bochner-Krall operators ᒁ with N ≤ 4. The corresponding BKS are various classical systems such as the Jacobi-type, the Laguerre-type, the Legendre-type, the Bessel and the Hermite polynomials, see [4], Th. 3.1.…”

Section: Introductionmentioning

confidence: 99%

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On Bochner-Krall orthogonal polynomial systems

Bergkvist¹,

Rullgård²,

Shapiro³

2004

MATH. SCAND.

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In this paper we address the classical question going back to S. Bochner and H. L. Krall to describe all systems {p n (x)} ∞ n=0 of orthogonal polynomials (OPS) which are the eigenfunctions of some finite order differential operator. SummaryConsider a sequence of polynomials {p n } ∞ n=0 in a variable x, where deg p n = n. This sequence is orthogonal with respect to a measure µ if p n (x)p m (x) dµ(x) is nonzero precisely when n = m. We are here concerned with polynomials orthogonal with respect to a measure of the so-called Nevai class, see [11] and below. Furthermore we say that {p n } is a sequence of eigenpolynomials if there exists a differential operator ᒁ = N k=1 a k (x) (d/dx) k where a k are polynomials in x. Finally, {p n } is called a Bochner-Krall system if it is both orthogonal and a system of eigenpolynomials. If it is orthogonal with respect to a measure of Nevai class, we say that it is a Bochner-Krall system of Nevai type.It is an open problem to classify all Bochner-Krall systems. In [4] it is conjectured that the leading coefficient a N for any Bochner-Krall system is a power of a polynomial of degree at most 2. Our main result is an affirmative answer to this conjecture for Bochner-Krall systems of Nevai type.

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“…In fact, L is an infinite order differential operator up to α ∈ N. In such a case, the order of the differential operator L is 2α + 4. For a general and recent survey about orthogonal polynomials as eigenfunctions of finite order differential operators, see the excellent review [5] by W. N. Everitt et al…”

Section: Introductionmentioning

confidence: 99%

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García−Caballero¹,

Moreno²,

Marcellán³

2003

Periodica Mathematica Hungarica

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In this contribution we analyze the generating functions for polynomials orthogonal with respect to a symmetric linear functional u, i.e., a linear application in the linear space of polynomials with complex coefficients such that u(x 2n+1 ) = 0. In some cases we can deduce explicitly the expression for the generating functionwhere {Pn}n is the sequence of orthogonal polynomials with respect to u.

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Orthogonal polynomial solutions of linear ordinary differential equations

Cited by 62 publications

References 63 publications

Askey-Wilson type functions with bound states

Askey-Wilson type functions with bound states

On Bochner-Krall orthogonal polynomial systems

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