Estimates of the Hermite and the Freud polynomials

Bonan, Stanford S.; Clark, Dean S.

doi:10.1016/0021-9045(90)90104-x

Cited by 103 publications

(77 citation statements)

References 5 publications

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“…Under the latter assumptions the orthonormal polynomials p n 's satisfy the differential recurrence relation [3], [4], [5], (1.8) and the differential second order equation…”

Section: Introductionmentioning

confidence: 99%

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An electrostatics model for zeros of general orthogonal polynomials

Ismail¹

2000

Pacific J. Math.

103

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We prove that the zeros of general orthogonal polynomials, subject to certain integrability conditions on their weight functions determine the equilibrium position of movable n unit charges in an external field determined by the weight function. We compute the total energy of the system in terms of the recursion coefficients of the orthonormal polynomials and study its limiting behavior as the number of particles tends to infinity in the case of Freud exponential weights.

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“…Under the latter assumptions the orthonormal polynomials p n 's satisfy the differential recurrence relation [3], [4], [5], (1.8) and the differential second order equation…”

Section: Introductionmentioning

confidence: 99%

“…The representations (1.5) and (1.6) of the A n 's and B n 's are in [5] but earlier versions are in [4] and [3].…”

Section: Introductionmentioning

confidence: 99%

An electrostatics model for zeros of general orthogonal polynomials

Ismail¹

2000

Pacific J. Math.

103

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“…There is a difficulty in extending the results of [ 1 ] in this way. Namely, this paper generalizes the results of [3] to asymmetric Freud weights, but for these weights a recurrence formula of the type (17) is not known in the case p / 0.…”

Section: Z=0 7=0 V+j=l (Mod R)mentioning

confidence: 64%

“…We obtained an asymptotic expansion for an in terms of powers of l/n in [8, pp. 427-428] [3], [12], and [13] appear to generalize in a similar fashion. There is a difficulty in extending the results of [ 1 ] in this way.…”

Section: Z=0 7=0 V+j=l (Mod R)mentioning

confidence: 77%

Asymptotic expansions for solutions of smooth recurrence equations

Jha¹,

Mate²,

Nevai³

1990

Proc. Amer. Math. Soc.

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Abstract.Let (y" : n > 1) be a convergent sequence of reals, where for each n the tuple (yn, yn+x, ... , yn+ k, X/n) satisfies one of r equations, depending on the residue class of n (mod r) , for some given k and r . Assume these equations are smooth, they have the same gradient in the first k + 1 variables, and this gradient satisfies a certain nonmodularity condition. We then show that y" has r asymptotic expansions, depending on the residue class of n (mod r), in terms of powers of 1 fn . This result enables us to discuss the asymptotic behavior of the recurrence coefficients associated with certain orthogonal polynomials. A key ingredient in the proof of the main result is a lemma involving exponential sums.

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“…The convention is taken that sums with upper limits less than their lower limits are zero, or equivalently coefficients with negative indices are zero. Contrary to appearances, the highest coefficient in (6.5) will turn out to be c q,p as c q,p+2 occurs with c 0,0 as a factor and c q,p+1 has a factor of 6) which is zero as a consequence of c 0,0 = 0 and (6.2). We find that the coefficient c q,p occurs as a linear term and has a factor of…”

Section: The σ -Function Expansion About S =mentioning

confidence: 90%

The Distribution of the First Eigenvalue Spacing at the Hard Edge of the Laguerre Unitary Ensemble

Forrester¹,

Witte²

2007

Kyushu J. Math.

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Abstract. The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite rank random matrices is found in terms of a Painlevé V system, and the solution of its associated linear isomonodromic system. In particular, it is characterized by the polynomial solutions to the isomonodromic equations which are also orthogonal with respect to a deformation of the Laguerre weight. In the scaling to the hard edge regime we find an analogous situation where a certain Painlevé III system and its associated linear isomonodromic system characterize the scaled distribution. We undertake extensive analytical studies of this system and use this knowledge to accurately compute the distribution and its moments for various values of the parameter a. In particular, choosing a = ± 1 2 allows the first eigenvalue spacing distribution for random real orthogonal matrices to be computed.

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Estimates of the Hermite and the Freud polynomials

Cited by 103 publications

References 5 publications

An electrostatics model for zeros of general orthogonal polynomials

An electrostatics model for zeros of general orthogonal polynomials

Asymptotic expansions for solutions of smooth recurrence equations

The Distribution of the First Eigenvalue Spacing at the Hard Edge of the Laguerre Unitary Ensemble

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