Extensions of umbral calculus II: double delta operators, Leibniz extensions and Hattori-Stong theorems

Francis Clarke[1]; John Hunton[2]; Nigel Ray[3]

  • [1] University of Wales Swansea, Department of Mathematics, Singleton Park, Swansea SA2 8PP (Grande-Bretagne)
  • [2] University of Leicester, Department of Mathematics and Computer Science, University Road, Leicester LE1 7RH (Grande-Bretagne)
  • [3] University of Manchester, Department of Mathematics, Oxford Road, Manchester M13 9PL (Grande-Bretagne)

Annales de l’institut Fourier (2001)

  • Volume: 51, Issue: 2, page 297-336
  • ISSN: 0373-0956

Abstract

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We continue our programme of extending the Roman-Rota umbral calculus to the setting of delta operators over a graded ring E * with a view to applications in algebraic topology and the theory of formal group laws. We concentrate on the situation where E * is free of additive torsion, in which context the central issues are number- theoretic questions of divisibility. We study polynomial algebras which admit the action of two delta operators linked by an invertible power series, and make related constructions motivated by the Hattori-Stong theorem of algebraic topology. Our treatment is couched purely in terms of the umbral calculus, but inspires novel topological applications. In particular we obtain a generalised form of the Hattori-Stong theorem.

How to cite

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Clarke, Francis, Hunton, John, and Ray, Nigel. "Extensions of umbral calculus II: double delta operators, Leibniz extensions and Hattori-Stong theorems." Annales de l’institut Fourier 51.2 (2001): 297-336. <http://eudml.org/doc/115917>.

@article{Clarke2001,
abstract = {We continue our programme of extending the Roman-Rota umbral calculus to the setting of delta operators over a graded ring $E_\{*\}$ with a view to applications in algebraic topology and the theory of formal group laws. We concentrate on the situation where $E_\{*\}$ is free of additive torsion, in which context the central issues are number- theoretic questions of divisibility. We study polynomial algebras which admit the action of two delta operators linked by an invertible power series, and make related constructions motivated by the Hattori-Stong theorem of algebraic topology. Our treatment is couched purely in terms of the umbral calculus, but inspires novel topological applications. In particular we obtain a generalised form of the Hattori-Stong theorem.},
affiliation = {University of Wales Swansea, Department of Mathematics, Singleton Park, Swansea SA2 8PP (Grande-Bretagne); University of Leicester, Department of Mathematics and Computer Science, University Road, Leicester LE1 7RH (Grande-Bretagne); University of Manchester, Department of Mathematics, Oxford Road, Manchester M13 9PL (Grande-Bretagne)},
author = {Clarke, Francis, Hunton, John, Ray, Nigel},
journal = {Annales de l’institut Fourier},
keywords = {umbral calculus; Hattori-Stong theorems; polynomial algebras; delta operators; power series; algebraic topology},
language = {eng},
number = {2},
pages = {297-336},
publisher = {Association des Annales de l'Institut Fourier},
title = {Extensions of umbral calculus II: double delta operators, Leibniz extensions and Hattori-Stong theorems},
url = {http://eudml.org/doc/115917},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Clarke, Francis
AU - Hunton, John
AU - Ray, Nigel
TI - Extensions of umbral calculus II: double delta operators, Leibniz extensions and Hattori-Stong theorems
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 2
SP - 297
EP - 336
AB - We continue our programme of extending the Roman-Rota umbral calculus to the setting of delta operators over a graded ring $E_{*}$ with a view to applications in algebraic topology and the theory of formal group laws. We concentrate on the situation where $E_{*}$ is free of additive torsion, in which context the central issues are number- theoretic questions of divisibility. We study polynomial algebras which admit the action of two delta operators linked by an invertible power series, and make related constructions motivated by the Hattori-Stong theorem of algebraic topology. Our treatment is couched purely in terms of the umbral calculus, but inspires novel topological applications. In particular we obtain a generalised form of the Hattori-Stong theorem.
LA - eng
KW - umbral calculus; Hattori-Stong theorems; polynomial algebras; delta operators; power series; algebraic topology
UR - http://eudml.org/doc/115917
ER -

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