Operational rules and a generalized Hermite polynomials

Chaggara, Hamza

doi:10.1016/j.jmaa.2006.09.068

Cited by 12 publications

(11 citation statements)

References 24 publications

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“…Clearly, the operator H α,q raises the degree of any polynomial. Such operator is called raising operator [9,13,19]. By transposition of the operator H α,q , we get…”

Section: Orthogonal Polynomials Via Raising Operator X − αH Qmentioning

confidence: 99%

HAHN'S PROBLEM WITH RESPECT TO SOME PERTURBATIONS OF THE RAISING OPERATOR \(X-c\)

Aloui

Souissi

2020

UMJ

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In this paper, we study the Hahn's problem with respect to some raising operators perturbed of the operator \(X-c\), where \(c\) is an arbitrary complex number. More precisely, the two following characterizations hold: up to a normalization, the \(q\)-Hermite (resp. Charlier) polynomial is the only \(H_{\alpha,q}\)-classical (resp. \(\mathcal{S}_{\lambda}\)-classical) orthogonal polynomial, where \(H_{\alpha, q}:=X+\alpha H_q\) and \(\mathcal{S}_{\lambda}:=(X+1)-\lambda\tau_{-1}.\)

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“…Clearly, the operator H α,q raises the degree of any polynomial. Such operator is called raising operator [9,13,19]. By transposition of the operator H α,q , we get…”

Section: Orthogonal Polynomials Via Raising Operator X − αH Qmentioning

confidence: 99%

HAHN'S PROBLEM WITH RESPECT TO SOME PERTURBATIONS OF THE RAISING OPERATOR \(X-c\)

Aloui

Souissi

2020

UMJ

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“…The general linearization problem consists in finding the coefficients B n ( j , k ) such that

Q_{j} false(x false) R_{k} false(x false) = \sum_{n = 0}^{j + k} B_{n} false(j, k false) P_{n} false(x false) .

A particular case of this problem is the standard linearization problem (or Clebsch–Gordan‐type problem) for which Q n ≡ R n ≡ P n :

P_{j} false(x false) P_{k} false(x false) = \sum_{n = 0}^{j + k} B_{n} false(j, k false) P_{n} false(x false) .

Another particular case is the so‐called connection problem, for which R k ( x ) = 1; if in addition Q j ( x ) = x j is chosen, it is referred to as the inversion problem for the sequence

{false{P_{n} false}}_{n ⩾ 0}

. These problems have been the subject of numerous investigations, some of them being addressed in and references therein. The objective of this communication is definitely not to review exhaustively these results but to apply them to the computation of higher‐order moments from PC or gPC expansions of random parameters and/or functionals.…”

Section: Standard Linearization Problemmentioning

confidence: 99%

“…This family reduces to the classical Laguerre polynomials

{false{L_{n} false}}_{n ⩾ 0}

for α = 0, applicable to exponentially distributed random variables. The normalization constant γ n in reads

γ_{n} = \frac{(n + α)!}{n!} .

The linearization coefficients B n ( j , k ) in the general linearization problem

L_{j}^{(λ)} (x) L_{k}^{(μ)} (x) = \sum_{n = | j - k |}^{j + k} B_{n} (j, k) L_{n}^{(α)} (x), α, λ, μ > - 1,

are given in [, equation (3.24)] in terms of double hypergeometric functions. Again, in the context of PC expansions, we are rather interested in the standard linearization problem λ = μ = α , for which the linearization coefficients are given by in terms of a terminating hypergeometric series 3 F 2 :

B_{j + k - n} (j, k) = \frac{{(- 2)}^{n}}{n!} \frac{(j + k - n)!}{(j - n)! (k - n)!}_{3} F_{2} (\begin{matrix} - \frac{n}{2}, - \frac{n - 1}{2}, j + \end{matrix}|)

…”

Section: Higher‐order Moments Of Orthonormal Polynomialsmentioning

confidence: 99%

Computation of higher‐order moments of generalized polynomial chaos expansions

Savin

Faverjon

2017

Numerical Meth Engineering

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Summary Because of the complexity of fluid flow solvers, non‐intrusive uncertainty quantification techniques have been developed in aerodynamic simulations in order to compute the quantities of interest required in an optimization process, for example. The objective function is commonly expressed in terms of moments of these quantities, such as the mean, standard deviation, or even higher‐order moments. Polynomial surrogate models based on polynomial chaos expansions have often been implemented in this respect. The original approach of uncertainty quantification using polynomial chaos is however intrusive. It is based on a Galerkin‐type formulation of the model equations to derive the governing equations for the polynomial expansion coefficients. Third‐order, indeed fourth‐order moments of the polynomials are needed in this analysis. Besides, both intrusive and non‐intrusive approaches call for their computation provided that higher‐order moments of the quantities of interest need to be post‐processed. In most applications, they are evaluated by Gauss quadratures and eventually stored for use throughout the computations. In this paper, analytical formulas are rather considered for the moments of the continuous polynomials of the Askey scheme, so that they can be evaluated by quadrature‐free procedures instead. Matlab© codes have been developed for this purpose and tested by comparisons with Gauss quadratures. Copyright © 2017 John Wiley & Sons, Ltd.

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“…Since the works of Ferrers and Adams in the late 1870s and Bailey, Dougall and Erdélyi in the early 1900s about the linearization formula of the product of two hypergeometric orthogonal polynomials (HOPs) of Legendre [1][2][3], Gegenbauer or ultraspherical [4] and Laguerre [5] type up until now, an intense activity about the linearization problem has been developed in the theory of HOPs from both theoretical [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and applied [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] standpoints. See also the monographs of Askey [36], Andrews-Askey-Roy [37], Ismail [38], Koekoek-Lesky-Swarttouw [39], Gautschi [26,35] and Tcheutia [32] for partial periodic reviews.…”

Section: Introductionmentioning

confidence: 99%

Linearization and Krein-like functionals of hypergeometric orthogonal polynomials

Dehesa

Moreno-Balcázar

Toranzo

2018

Journal of Mathematical Physics

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The Krein-like r-functionals of the hypergeometric orthogonal polynomials {pn(x)} with kernel of the form x s [ω(x)] β pm 1 (x) . . . pm r (x), being ω(x) the weight function on the interval ∆ ∈ R, are determined by means of the Srivastava linearization method. The particular 2-functionals, which are particularly relevant in quantum physics, are explicitly given in terms of the degrees and the characteristic parameters of the polynomials. They include the well-known power moments and the novel Krein-like moments. Moreover, various related types of exponential and logarithmic functionals are also investigated.

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Operational rules and a generalized Hermite polynomials

Abstract: In this paper, we use operational rules associated with three operators corresponding to a generalized Hermite polynomials introduced by Szegö to derive, as far as we know, new proofs of some known properties as well as new expansions formulae related to these polynomials.

Cited by 12 publications

References 24 publications

HAHN'S PROBLEM WITH RESPECT TO SOME PERTURBATIONS OF THE RAISING OPERATOR \(X-c\)

HAHN'S PROBLEM WITH RESPECT TO SOME PERTURBATIONS OF THE RAISING OPERATOR \(X-c\)

Computation of higher‐order moments of generalized polynomial chaos expansions

Linearization and Krein-like functionals of hypergeometric orthogonal polynomials

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