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Hermite and Gegenbauer polynomials in superspace using Clifford analysis

H De Bie and F Sommen

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The Clifford–Hermite and the Clifford–Gegenbauer polynomials of standard Clifford analysis are generalized to the new framework of Clifford analysis in superspace in a merely symbolic way. This means that one does not a priori need an integration theory in superspace. Furthermore, a lot of basic properties, such as orthogonality relations, differential equations and recursion formulae, are proven. Finally, an interesting physical application of the super Clifford–Hermite polynomials is discussed, thus giving an interpretation to the super-dimension.


PACS

02.10.De Algebraic structures and number theory

02.30.Tb Operator theory

MSC

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (See also 42C05 for general orthogonal polynomials and functions)

34L40 Particular operators (Dirac, one-dimensional Schrödinger, etc.)

15A66 Clifford algebras, spinors

Subjects

Mathematical physics

Dates

Issue 34 (24 August 2007)

Received 12 June 2007

Published 7 August 2007



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