Corrigendum to: “The calculation of multidimensional Hermite polynomials and Gram-Charlier coefficients” (Math. Comp. 24 (1970), 537–545) by Berkowitz and F. J. Garner

Berkowitz, Sidney

doi:10.1090/s0025-5718-1971-0400655-1

Cited by 3 publications

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“…At the moment this strategy seems to give the best results in terms of memory management, leading however to rather complex algorithms. It is worth noticing that Berkowitz' algorithm [28] for the computation of multidimensional Hermite polynomials allows to compute a polynomial of a given degree with the least number of recursions. Its modification for the calculation of FC integrals is actually under investigation.…”

Section: Duschinsky's Transformationmentioning

confidence: 99%

Franck–Condon factors—Computational approaches and recent developments

2013

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Algorithms and methodologies for the calculation of Franck–Condon integrals are reviewed. Starting from the standard approach based on the Cartesian representation of the normal modes and the use of Duschinsky's transformation and recursion formulas, methods for treating large displacements of the equilibrium positions and anharmonic and non-Condon effects are considered. Finally, focusing attention on problems arising from the use of recurrence relations, some of the proposed solutions are critically reviewed along with new alternative approaches.

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Section: Duschinsky's Transformationmentioning

confidence: 99%

Franck–Condon factors—Computational approaches and recent developments

2013

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“…In this subsection, we will revise the matter statistics for the one‐dimensional probability distribution function as developed in the works by Juszkiewicz et al (1995), Bernardeau & Kofman (1995) and Colombi (1994). For a general overview on asymptotic statistical techniques, see Berkowitz & Garner (1970) and Barndorff‐Nielsen & Cox (1989). Other univariate matter field distribution functions have been proposed.…”

Section: Gravitational Clustering Field Statisticsmentioning

confidence: 99%

“…Let us study here the multivariate Gram–Charlier series expansion (see Berkowitz & Garner 1970): with G ( ν ) being a multivariate Gaussian distribution .…”

Section: Gravitational Clustering Field Statisticsmentioning

confidence: 99%

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Non-Gaussian gravitational clustering field statistics

Kitaura

2012

Monthly Notices of the Royal Astronomical Society

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In this work we investigate the multivariate statistical description of the matter distribution in the nonlinear regime. We introduce the multivariate Edgeworth expansion of the lognormal distribution to model the cosmological matter field. Such a technique could be useful to generate and reconstruct three-dimensional nonlinear cosmological density fields with the information of higher order correlation functions. We explicitly calculate the expansion up to third order in perturbation theory making use of the multivariate Hermite polynomials up to sixth order. The probability distribution function for the matter field includes at this level the two-point, the three-point and the four-point correlation functions. We use the hierarchical model to formulate the higher order correlation functions based on combinations of the two-point correlation function. This permits us to find compact expressions for the skewness and kurtosis terms of the expanded lognormal field which can be efficiently computed. The method is, however, flexible to incorporate arbitrary higher order correlation functions which have analytical expressions. The applications of such a technique can be especially useful to perform weak-lensing or neutral hydrogen 21 cm line tomography, as well as to directly use the galaxy distribution or the Lyman-alpha forest to study structure formation.Comment: 20 pages, 2 figures; accepted in MNRAS 2011 August 22, in original form 2010 December 14 published, Publication Date: 03/201

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“…For the case of a gaussian weight he obtained symmetric tensorial polynomials and showed them to reduce to the well-known previously found one-dimensional Hermite polynomials by taking the limit D → 1. Since then the D-dimensional Hermite polynomials have been studied under several aspects, such as the obtainment of recurrence formulas 5 . Nevertheless the tensorial orthonormal polynomials have not been systematically studied for weights other than the gaussian one so far, apart from a few attempts, such as the Laguerre weight 6 in D=2 with the intent to apply in quantum optics 7,8 .…”

Section: Introductionmentioning

confidence: 99%

Chebyshev, Legendre, Hermite and Other Orthonormal Polynomials in D Dimensions

Doria

Coelho

2018

Reports on Mathematical Physics

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We propose a general method to construct symmetric tensor polynomials in the D-dimensional Euclidean space which are orthonormal under a general weight. The D-dimensional Hermite polynomials are a particular case of the present ones for the case of a gaussian weight. Hence we obtain generalizations of the Legendre and of the Chebyshev polynomials in D dimensions that reduce to the respective well-known orthonormal polynomials in D=1 dimensions. We also obtain new Ddimensional polynomials orthonormal under other weights, such as the Fermi-Dirac, Bose-Einstein, Graphene equilibrium distribution functions and the Yukawa potential. We calculate the series expansion of an arbitrary function in terms of the new polynomials up to the fourth order and define orthonormal multipoles. The explicit orthonormalization of the polynomials up to the fifth order (N from 0 to 4) reveals an increasing number of orthonormalization equations that matches exactly the number of polynomial coefficients indication the correctness of the present procedure.

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Corrigendum to: “The calculation of multidimensional Hermite polynomials and Gram-Charlier coefficients” (Math. Comp. 24 (1970), 537–545) by Berkowitz and F. J. Garner

Cited by 3 publications

References 0 publications

Franck–Condon factors—Computational approaches and recent developments

Franck–Condon factors—Computational approaches and recent developments

Non-Gaussian gravitational clustering field statistics

Chebyshev, Legendre, Hermite and Other Orthonormal Polynomials in D Dimensions

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