Extended finite operator calculus-an example of algebraization of analysis

Andrzej Kwaśniewski; Ewa Borak

Open Mathematics (2004)

  • Volume: 2, Issue: 5, page 767-792
  • ISSN: 2391-5455

Abstract

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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 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]).

How to cite

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Andrzej Kwaśniewski, and Ewa Borak. "Extended finite operator calculus-an example of algebraization of analysis." Open Mathematics 2.5 (2004): 767-792. <http://eudml.org/doc/268923>.

@article{AndrzejKwaśniewski2004,
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]).},
author = {Andrzej Kwaśniewski, Ewa Borak},
journal = {Open Mathematics},
keywords = {05A40; 81S99},
language = {eng},
number = {5},
pages = {767-792},
title = {Extended finite operator calculus-an example of algebraization of analysis},
url = {http://eudml.org/doc/268923},
volume = {2},
year = {2004},
}

TY - JOUR
AU - Andrzej Kwaśniewski
AU - Ewa Borak
TI - Extended finite operator calculus-an example of algebraization of analysis
JO - Open Mathematics
PY - 2004
VL - 2
IS - 5
SP - 767
EP - 792
AB - “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]).
LA - eng
KW - 05A40; 81S99
UR - http://eudml.org/doc/268923
ER -

References

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