Orthogonal polynomials and their derivatives, I

Bonan, Stanford S.; Nevai, Paul

doi:10.1016/0021-9045(84)90023-6

Cited by 76 publications

(54 citation statements)

References 7 publications

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“…Here we provide a brief guide to the literature on this subject. See, for example, [3,4,10,7,9,8,12,19,20,32,33,38]. In fact, Magnus in [33] noted that such operators were known to Laguerre.…”

Section: Ladder Operators and Non-linear Difference Equationsmentioning

confidence: 95%

Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I

Chen

Its

2010

Journal of Approximation Theory

132

193

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In this paper, we study a certain linear statistics of the unitary Laguerre ensembles, motivated in part by an integrable quantum field theory at finite temperature. It transpires that this is equivalent to the characterization of a sequence of polynomials orthogonal with respect to the weightnamely, the determination of the associated Hankel determinant and recurrence coefficients. Here w(x, s) is the Laguerre weight x α e −x perturbed by a multiplicative factor e −s/x , which induces an infinitely strong zero at the origin.For polynomials orthogonal on the unit circle, a particular example where there are explicit formulas, the weight of which has infinitely strong zeros, was investigated by Pollaczek and Szegö many years ago. Such weights are said to be singular or irregular due to the violation of the Szegö condition.In our problem, the linear statistics is a sum of the reciprocal of positive random variables {x j : j = 1, . . . , n}; n j=1 1/x j . We show that the moment generating function, or the Laplace transform of the probability density function of this linear statistics, can be expressed as the ratio of Hankel determinants and as an integral involving a particular third Painlevé function.

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Section: Ladder Operators and Non-linear Difference Equationsmentioning

confidence: 95%

Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I

Chen

Its

2010

Journal of Approximation Theory

132

193

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“…The study of orthogonal polynomials with respect to semiclassical functionals (2), (7), (5), and (6), started more than a hundred years ago with the work of Laguerre [17]. In spite of its long history and a number of powerful modern results (see, for example [18,24,26]) one cannot say that the theory of semiclassical OP enjoys the same level of development and completeness as the theory of classical polynomials.…”

Section: Semiclassical Orthogonal Polynomialsmentioning

confidence: 96%

“…As a matter of fact, it has been proven in [4,21,22] that the space of solutions of the problem (2), (7), and (5) has dimension s+1, which means that for each solution, w(z), of the differential equation (2), there exist s+1 homotopically different classes of paths of integration [# j ] s+1 j=1 , satisfying (5), such that pairs [w(z), # j ] s+1 j=1 generate, by means of (6), (s+1) linearly independent moment sequences.…”

Section: Semiclassical Orthogonal Polynomialsmentioning

confidence: 99%

Semiclassical Multiple Orthogonal Polynomials and the Properties of Jacobi–Bessel Polynomials

Aptekarev

Marcellán

Rocha

1997

Journal of Approximation Theory

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This paper deals with Hermite Pade polynomials in the case where the multiple orthogonality condition is related to semiclassical functionals. The polynomials, introduced in such a way, are a generalization of classical orthogonal polynomials (Jacobi, Laguerre, Hermite, and Bessel polynomials). They satisfy a Rodrigues type formula and an (s+2)-order differential equation, where s is the class of the semiclassical functional. A special case of polynomials, multiple orthogonal with respect to the semiclassical weight function w(x)=x : 0 (x&a) : 1 e #Âx (a combination of the classical weights of Jacobi and Bessel), is analyzed in order to obtain the strong (Szego type) asymptotics and the zero distribution.1997 Academic Press

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“…Among others it is proved that the derivatives of any orders of Freud orthonormal polynomials constitute Hilbertian bases in weighted spaces. Some further results concerning the derivatives of Freud-type polynomials can be found in [1], [2]. Some notations will be used.…”

Section: §1 Introductionmentioning

confidence: 99%

On biorthogonal systems associated to orthogonal polynomials

2010

Acta Math Hung

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We consider biorthogonal systems of functions associated to derivatives of orthogonal polynomials in the case of general weights. For Freud polynomials, it is proved that the derivatives of any orders of them constitute Hilbertian bases in the space of weighted square integrable functions. §1. IntroductionIt is known a characteristic property of the classical orthogonal polynomials, which states that their derivatives are also orthogonal polynomials. Such a property has many applications in the theory of orthogonal expansions and numerical computations of derivatives. Our purpose in the present paper is to deal with derivatives of non classical orthogonal polynomials from the aspect of expansions and approximations.In Section 2 we consider biorthogonal systems associated to derivatives of orthogonal polynomials in the cases of general weights. Section 3 deals with Freud weights. Among others it is proved that the derivatives of any orders of Freud orthonormal polynomials constitute Hilbertian bases in weighted spaces. Some further results concerning the derivatives of Freud-type polynomials can be found in [1], [2]. Some notations will be used. L p (a, b) (−∞ a < b ∞, 1 p ∞) denotes the space of all functions f defined on (a, b) with the norms

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Orthogonal polynomials and their derivatives, I

Cited by 76 publications

References 7 publications

Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I

Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I

Semiclassical Multiple Orthogonal Polynomials and the Properties of Jacobi–Bessel Polynomials

On biorthogonal systems associated to orthogonal polynomials

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