Displaying similar documents to “More on deformed oscillator algebras and extended umbral calculus”

Further remarks on extended umbral calculus

Kwaśniewski, A. K., Grądzka, E.

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The paper deals with extensions of the finite operator calculus of G.-C. Rota called ψ -extensions which give a framework for corresponding quantum group investigations. This also covers the instance of the well-known q -analogue of umbral calculus. The article also contains glossaries of the most important terms and notations used by Ward, Viskov, Markowsky and Roman on one side and the Rota-oriented notations on the other side.

Linearization of Arbitrary products of classical orthogonal polynomials

Mahouton Hounkonnou, Said Belmehdi, André Ronveaux (2000)

Applicationes Mathematicae

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A procedure is proposed in order to expand w = ∏ j = 1 N P i j ( x ) = ∑ k = 0 M L k P k ( x ) where P i ( x ) belongs to aclassical orthogonal polynomial sequence (Jacobi, Bessel, Laguerre and Hermite) ( M = ∑ j = 1 N i j ). We first derive a linear differential equation of order 2 N satisfied by w, fromwhich we deduce a recurrence relation in k for the linearizationcoefficients L k . We develop in detail the two cases [ P i ( x ) ] N , P i ( x ) P j ( x ) P k ( x ) and give the recurrencerelation in some cases (N=3,4), when the polynomials P i ( x ) are monic Hermite orthogonal polynomials.

Extended finite operator calculus-an example of algebraization of analysis

Andrzej Kwaśniewski, Ewa Borak (2004)

Open Mathematics

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“A Calculus of Sequences” started in 1936 by Ward constitutes the general scheme for extensions of classical operator calculus of Rota-Mullin considered by many afterwards and after Ward. Because of the notation we shall call the Ward's calculus of sequences in its afterwards elaborated form-a ψ-calculus. The ψ-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota-Mullin or equivalently-of umbral calculus of Roman and Rota. At the...