Heisenberg algebra, umbral calculus and orthogonal polynomials

Dattoli, G.; Levi, Decio; Winternitz, P.

doi:10.1063/1.2909731

Cited by 16 publications

(17 citation statements)

References 24 publications

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“…Because of this correspondence, we would like to stress that the quantum mechanical description of umbral calculus gives us many degrees of freedom to construct the raising operators in such way that the commuting relations 1 fulfill. In particular, in , it was pointed out the importance to consider the following symmetrized versions of as a special type of canonical discretization.…”

Section: Umbral Clifford Analysismentioning

confidence: 99%

(Discrete) Almansi type decompositions: an umbral calculus framework based on osp(1|2) symmetries

Faustino

Ren

2011

Math. Meth. Appl. Sci.

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Communicated by Svetlin GeorgievWe introduce the umbral calculus formalism for hypercomplex variables starting from the fact that the algebra of multivariate polynomials IROEx shall be described in terms of the generators of the Weyl-Heisenberg algebra. The extension of IROEx to the algebra of Clifford-valued polynomials P gives rise to an algebra of Clifford-valued operators whose canonical generators are isomorphic to the orthosymplectic Lie algebra osp.1j2/.This extension provides an effective framework in continuity and discreteness that allow us to establish an alternative formulation of

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Section: Umbral Clifford Analysismentioning

confidence: 99%

(Discrete) Almansi type decompositions: an umbral calculus framework based on osp(1|2) symmetries

Faustino

Ren

2011

Math. Meth. Appl. Sci.

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“…In a numbers of previous papers [5,6,7,8] it has been established that the umbral image of a Bessel function is a Gaussian. This statement can be profitably exploited within the present context, provided we premise some concepts [8,9].…”

Section: Umbral Methods and The Negative Derivative Formalismmentioning

confidence: 99%

Operational Versus Umbral Methods and the Borel Transform

Dattoli

Palma

Sabia

et al. 2017

Int. J. Appl. Comput. Math

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Abstract. Integro-differential methods, currently exploited in calculus, provide an inexhaustible source of tools to be applied to a wide class of problems, involving the theory of special functions and other subjects. The use of integral transforms of the Borel type and the associated formalism is shown to be a very effective mean, constituting a solid bridge between umbral and operational methods. We merge these different points of view to obtain new and efficient analytical techniques for the derivation of integrals of special functions and the summation of associated generating functions as well.

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“…Methods of algebraic and umbral nature possess, among the other advantages, the undoubtful merit of simplifying practical computational issues. In a numbers of previous papers [1][2][3][4][5][6][7] it has been established that the one of the possible umbral images of a Bessel function is a Gaussian. This statement can be profitably exploited in the present context, provided we establish some reference tools [6,8].…”

Section: Introductionmentioning

confidence: 99%