Extended finite operator calculus—an example of algebraization of analysis |
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Authors: | Andrzej Krzysztof Kwaśniewski Ewa Borak |
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Institution: | (1) Higher School of Mathematics and Applied Informatics, ul.Kamienna 17, 15-021 Bialystok, Poland;(2) Institute of Computer Science, Białystok University, ul.Sosnowa 64, 15-887 Białystok, Poland |
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Abstract: | “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 same time this calculus is an example of the algebraization of the analysis—here restricted to the algebra of polynomials.
Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus where Q stands for a generalized difference operator, i.e. the one lowering the degree of any polynomial by one.
This is a review article based on the recent first author contributions 1]. As the survey article it is supplemented by the
short indicatory glossaries of notation and terms used by Ward 2], Viskov 7, 8], Markowsky 12], Roman 28–32] on one side
and the Rota-oriented notation on the other side 9–11, 1, 3, 4, 35] (see also 33]). |
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Keywords: | Extended umbral calculus Graves-Heisenberg-Weyl algebra |
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