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Umbral Calculus

Formal power series , トポロジ でも である えば , complex oriented cohomology theory する formal group law として れる

その のことを しているときに , Ray [ Ray87 ] umbral calculus というもの Umbral calculus るには , Di Bucchianico Loeb survey [ DBL95 ] るのが , 9 しかないが , 506 もの げているので 34 にな ている

それによると , umbral calculus とは x 0 ,x 1 , ,x n , p 0 ,p 1 , ,p n , 研究 である

えば , ( x ) n = x ( x - 1) ⋅⋅⋅ ( x - n + 1) については 2 , Taylor つことが られている ただし , Taylor では 微分 -d
dx forward difference operator (Δ f )( x ) = f ( x + 1) - f ( x ) えないといけな いが

としては , Roman [ Rom84 ] Roman Rota [ RR78 ] などがある

formal power series していることは , umbral calculus 使 えると てよいようである Di Bucchianico Loeb survey [ DBL95 ] には のような げられている

にも numerical analysis, , 率論 , invariant theory などへの もあるら しい

また , survey ていないこととして のようなものもある

Di Bucchianico Loeb survey かれている

  • Sheffer polynomial から generalized Appell polynomial (Boas-Buck polynomial) への (Viskov [ Vis75 ] )
  • entire function への (Grabiner [ Gra88 Gra89 ] )
  • , そして への (Di Bucchianico, Loeb, Rota [ DBLR98 ] )
  • x - 1 log x への (Loeb [ Loe89 LR89 Loe91 ] )
  • linear operator への (Kurbanov Maksimov [ KM86 ] )
  • への (Van Hamme [ VH92 ] Verdoodt [ Ver96 Ver98 ] )

References

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[Cho88]     Frank M. Cholewinski. The finite calculus associated with Bessel functions , volume 75 of Contemporary Mathematics . American Mathematical Society, Providence, RI, 1988, https://doi.org/10.1090/conm/075 .

[DBL95]     A. Di Bucchianico and D. Loeb. A selected survey of umbral calculus. Electron. J. Combin. , 2:Dynamic Survey 3, 28, 1995, http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS3 .

[DBLR98]     Alessandro Di Bucchianico, Daniel E. Loeb, and Gian-Carlo Rota. Umbral calculus in Hilbert space. In Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996) , volume 161 of Progr. Math. , pages 213–238. Birkhäuser Boston, Boston, MA, 1998.

[Gra88]     Sandy Grabiner. Convergent expansions and bounded operators in the umbral calculus. Adv. in Math. , 72(1):132–167, 1988, https://doi.org/10.1016/0001-8708(88)90020-5 .

[Gra89]     Sandy Grabiner. Using Banach algebras to do analysis with the umbral calculus. In Conference on Automatic Continuity and Banach Algebras (Canberra, 1989) , volume 21 of Proc. Centre Math. Anal. Austral. Nat. Univ. , pages 170–185. Austral. Nat. Univ., Canberra, 1989.

[JR79]     S. A. Joni and G.-C. Rota. Coalgebras and bialgebras in combinatorics. Stud. Appl. Math. , 61(2):93–139, 1979.

[Kho90]     Alexander Nickolaevich Kholodov. The umbral calculus and orthogonal polynomials. Acta Appl. Math. , 19(1):1–54, 1990.

[KM86]     S. G. Kurbanov and V. M. Maksimov. Mutual decompositions of differential operators and operators of divided difference. Dokl. Akad. Nauk UzSSR , (4):8–9, 1986.

[Loe89]     Daniel Elliott Loeb. The iterated logarithmic algebra . ProQuest LLC, Ann Arbor, MI, 1989. Thesis (Ph.D.)–Massachusetts Institute of Technology.

[Loe90]     Daniel E. Loeb. Sequences of symmetric functions of binomial type. Stud. Appl. Math. , 83(1):1–30, 1990, https://doi.org/10.1002/sapm19908311 .

[Loe91]     Daniel E. Loeb. The iterated logarithmic algebra. Adv. Math. , 86(2):155–234, 1991, https://doi.org/10.1016/0001-8708(91)90041-5 .

[LR89]     Daniel E. Loeb and Gian-Carlo Rota. Formal power series of logarithmic type. Adv. Math. , 75(1):1–118, 1989, https://doi.org/10.1016/0001-8708(89)90079-0 .

[Mei34]     J. Meixner. Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion. J. London Math. Soc. , S1-9(1):6–13, 1934, https://doi.org/10.1112/jlms/s1-9.1.6 .

[MR70]     Ronald Mullin and Gian-Carlo Rota. On the foundations of combinatorial theory. III. Theory of binomial enumeration. In Graph Theory and its Applications (Proc. Advanced Sem., Math. Research Center, Univ. of Wisconsin, Madison, Wis., 1969) , pages 167–213 (loose errata). Academic Press, New York, 1970.

[NS82]     Warren Nichols and Moss Sweedler. Hopf algebras and combinatorics. In Umbral calculus and Hopf algebras (Norman, Okla., 1978) , volume 6 of Contemp. Math. , pages 49–84. Amer. Math. Soc., Providence, R.I., 1982.

[Ray87]     Nigel Ray. Symbolic calculus: a 19th century approach to M U and BP. In Homotopy theory (Durham, 1985) , volume 117 of London Math. Soc. Lecture Note Ser. , pages 195–238. Cambridge Univ. Press, Cambridge, 1987.

[Ray88]     Nigel Ray. Umbral calculus, binomial enumeration and chromatic polynomials. Trans. Amer. Math. Soc. , 309(1):191–213, 1988, http://dx.doi.org/10.2307/2001165 .

[Rob]     Thomas J. Robinson. Formal calculus and umbral calculus, arXiv:0912.0961 .

[Rom84]     Steven Roman. The umbral calculus , volume 111 of Pure and Applied Mathematics . Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984.

[RR78]     Steven M. Roman and Gian-Carlo Rota. The umbral calculus. Advances in Math. , 27(2):95–188, 1978.

[RW88]     Nigel Ray and Colin Wright. Colourings and partition types: a generalised chromatic polynomial. Ars Combin. , 25(B):277–286, 1988. Eleventh British Combinatorial Conference (London, 1987).

[SVY97]     D. Senato, A. Venezia, and J. Yang. Möbius polynomial species. Discrete Math. , 173(1-3):229–256, 1997, https://doi.org/10.1016/S0012-365X(96)00133-1 .

[Uen88]     Kazuo Ueno. Umbral calculus and special functions. Adv. in Math. , 67(2):174–229, 1988, https://doi.org/10.1016/0001-8708(88)90040-0 .

[Uen90]     Kazuo Ueno. Hypergeometric series formulas through operator calculus. Funkcial. Ekvac. , 33(3):493–518, 1990, http://www.math.kobe-u.ac.jp/ ~ fe/xml/mr1086774.xml .

[Ver96]     Ann Verdoodt. p -adic q -umbral calculus. J. Math. Anal. Appl. , 198(1):166–177, 1996, https://doi.org/10.1006/jmaa.1996.0074 .

[Ver98]     Ann Verdoodt. Non-Archimedean umbral calculus. Ann. Math. Blaise Pascal , 5(1):55–73, 1998, http://www.numdam.org/item?id=AMBP_1998__5_1_55_0 .

[VH92]     Lucien Van Hamme. Continuous operators which commute with translations, on the space of continuous functions on Z p . In p -adic functional analysis (Laredo, 1990) , volume 137 of Lecture Notes in Pure and Appl. Math. , pages 75–88. Dekker, New York, 1992.

[Vis75]     O. V. Viskov. Operator characterization of generalized Appell polynomials. Dokl. Akad. Nauk SSSR , 225(4):749–752, 1975.

[WR86]     Brian G. Wilson and Forrest J. Rogers. Umbral calculus and the theory of multispecies nonideal gases. Phys. A , 139(2-3):359–386, 1986, https://doi.org/10.1016/0378-4371(86)90126-3 .