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2004 | 2 | 5 | 767-792
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Extended finite operator calculus-an example of algebraization of analysis

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EN
“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]).
Wydawca
Czasopismo
Rocznik
Tom
2
Numer
5
Strony
767-792
Opis fizyczny
Daty
wydano
2004-10-01
online
2004-10-01
Twórcy
  • Higher School of Mathematics and Applied Informatics, kwandr@pl
autor
  • Białystok University
Bibliografia
  • [1] A.K. Kwaśniewski: “On Simple Characterisations of Sheffer ψ-polynomials and Related Propositions of the Calculus of Sequences”, Bulletin de la Soc. des Sciences et des Letters de Łódź 52 SERIE Reserchers sur les deformations, Vol. 36(45), (2002) (ArXiv: math.CO/0312397).
  • [2] M. Ward: “A Calculus of Sequences”, Amer. J. Math., Vol. 58(255), (1936).
  • [3] A.K. Kwaśniewski: “Towards ψ-extension of Finite Operator Calculus of Rota”, Rep. Math. Phys., Vol. 48(3, (2001) (ArXiv: math.CO/0402078 Feb 2004).
  • [4] A.K. Kwaśniewski: “On extended finite operator calculus of Rota and quantum groups”, Integral Transforms and Special Functions, Vol. 2(4), (2001).
  • [5] R.P. Boas and R.C. Buck Jr.: “Polynomials Defined by Generating Relations”, Am. Math. Monthly, Vol. 63(626), (1959).
  • [6] R.P. Boas and R.C. Buck Jr.: Polynomial Expansions of Analytic Functions, Springer, Berlin, 1964.
  • [7] O.V. Viskov: “Operator characterization of generalized Appel polynomials”, Soviet Math. Dokl., Vol. 6(1521), (1975).
  • [8] O.V. Viskov: “On the basis in the space of polynomials”, Soviet Math. Dokl., Vol. 19(250), (1978).
  • [9] G.-C. Rota and R. Mullin: On the foundations of combinatorial theory, III. Theory of Binomial Enumeration in “Graph Theory and Its Applications”, Academic Press, New York, 1970.
  • [10] G.C. Rota, D. Kahaner and A. Odlyzko: “On the Foundations of combinatorial theory. VIII. Finite operator calculus”, J. Math. Anal. Appl., Vol. 42(684), (1973).
  • [11] G.C. Rota: Finite Operator Calculus, Academic Press, New York, 1975.
  • [12] G. Markowsky: “Differential operators and the Theory of Binomial Enumeration”, J. Math. Anal. Appl., Vol. 63(145), (1978).
  • [13] A.K. Kwaśniewski: “Higher Order Recurrences for Analytical Functions of Tchebysheff Type”, Advances in Applied Clifford Algebras, Vol. 9(41), (1999).
  • [14] O.V. Viskov: “Noncommutative Approach to Classical Problems of Analysis”, Trudy Matiematicz'eskovo Instituta AN SSSR, Vol. 177(21), (1986).
  • [15] A. Di Bucchianico and D. Loeb: “A Simpler Characterization of Sheffer Polynomials, Studies in Applied Mathematics”, J. Math. Anal. Appl., Vol. 92(1), (1994).
  • [16] N.Ya. Sonin: “Rjady Ivana Bernulli”, Izw. Akad. Nauk, Vol. 7(337), (1897).
  • [17] C. Graves: “On the principles which regulate the interhange of symbols in certain symbolic equations”, Proc. Royal Irish Academy, Vol. 6(144), (1853–1857).
  • [18] P. Feinsilver and R. Schott: Algebraic Structures and Operator Calculus, Kluwer Academic Publishers, New York, 1993.
  • [19] O.V. Viskov: “Newton-Leibnitz Formula and the Taylor Expansion”, Integral Transforms and Special Functions, Vol. 12), (1997).
  • [20] S. Pincherle and U. Amaldi: Le operazioni distributive e le loro applicazioni all'analisi, N. Zanichelli, Bologna, 1901.
  • [21] S.G. Kurbanov and V.M. Maximov: “Mutual Expansions of Differential operators and Divided Difference Operators”, Dokl. Akad. Nauk Uz. SSSR, Vol. 4(8), (1986).
  • [22] A. Di Bucchianico and D. Loeb: Integral Transforms and Special Functions, Vol. 4(49), (1996).
  • [23] P. Kirschenhofer: “Binomialfolgen, Shefferfolgen und Faktorfolgen in der q-Analysis”, Sitzunber. Abt. II Oster. Ackad. Wiss. Math. Naturw. Kl., Vol. 188(263), (1979).
  • [24] A. Di Bucchianico and D. Loeb: “Sequences of Binomial Type with persistent roots”, J. Math. Anal. Appl., Vol. 199(39), (1996).
  • [25] F.H. Jackson: “q-difference equations”, Quart. J. Pure and Appl. Math., Vol. 41(193), (1910).
  • [26] F.H. Jackson: “The q-integral analogous to Borels integral”, Messenger of Math., Vol. 47(57), (1917).
  • [27] F.H. Jackson: “Basic Integration”, Quart. J. Math., Vol. 2(1), (1951).
  • [28] S.M. Roman: “The Theory of Umbral Calculus. I”, J. Math. Anal. Appl., Vol. 87(58), (1982).
  • [29] S.M. Roman: “The Theory of Umbral Calculus. II”, J. Math. Anal. Appl., Vol. 89(290), (1982).
  • [30] S.M. Roman: “The Theory of Umbral Calculus. III”, J. Math. Anal. Appl., Vol. 95(528), (1983).
  • [31] S.M. Roman: The umbral calculus, Academic Press, New York, 1984.
  • [32] S.R. Roman: “More on the umbral calculus with emphasis on the q-umbral calculus”, J. Math. Anal. Appl., Vol. 107(222), (1985).
  • [33] A.K. Kwasniewski and E. Gradzka: “Further remarks on ψ-extensions of finite operator calculus”, Rendiconti del Circolo Matematico di Palermo Serie II, Suppl., 69(117), (2002).
  • [34] J.F. Steffensen: “The poweroid an extension of the mathematical notion of power”, Acta Mathematica, Vol. 73(333), (1941).
  • [35] A.K. Kwaśniewski: “Main theorems of extended finite operator calculus”, Integral Transforms and Special Functions, Vol. 14(499), (2003).
  • [36] A.K. Kwaśniewski: “The logarithmic Fib-binomial formula”, Advan. Stud. Contemp. Math., Vol. 9(1), (2004) (19–26 ArXiv: math.CO/0406258 13 June 2004).
  • [37] A.K. Kwaśniewski: “On ψ-basic Bernoulli-Ward polynomials”, Bull. Soc. Sci. Lett. Lodz, in print (ArXiv: math.CO/0405577 30 May 2004).
  • [38] A.K. Kwaśniewski: “ψ-Appell polynomials' solutions of the-difference calculus nonhomogeneous equation”, Bull. Soc. Sci. Lett. Lodz, in print (ArXiv: math. CO/0405578 30 May 2004).
  • [39] A.K. Kwaśniewski: “On ψ-umbral difference Bernoulli-Taylor formula with Cauchy type remainder”, Bull. Soc. Sci. Lett. Lodz, in print (ArXiv: math.GM/0312401 December 2003).
  • [40] A.K. Kwaśniewski: “First contact remarks on umbra difference calculus references streams”, Bull. Soc. Sci. Lett. Lodz, in print (ArXiv: math.CO/0403139 v1 8 March 2004).
  • [41] A.K. Kwaśniewski: “On extended umbral calculus, oscillator-like algebras and Generalized Clifford Algebra”, Advances in Applied Clifford Algebras, Vol. 11(2), (2001), pp. 267–279 (ArXiv: math.QA/0401083 January 2004).
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_BF02475976
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