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

Clarke, Francis; Hunton, John; Ray, Nigel

Annales de l'Institut Fourier, Tome 51 (2001) no. 2, pp. 297-336.

Nous poursuivons notre projet en vue d’étendre le calcul “umbral” de Roman-Rota au contexte des opérateurs “delta” sur un anneau gradué $E_{*}$ dans le but de développer des applications en topologie algébrique et en théorie des lois de groupes formels. Nous visons la situation où $E_{*}$ est libre de torsion additive; dans cette situation les questions centrales sont celles de la divisibilité. Nous étudions les algèbres de polynômes qui admettent l’action de deux opérateurs “delta” liés par une série inversible, et nous proposons des constructions connexes motivées par le théorème de Hattori-Stong en topologie algébrique. Notre traitement se poursuit exclusivement en termes de calcul “umbral”, ce qui nous mène à des applications topologiques nouvelles. En particulier, nous arrivons à une forme généralisée du théorème de Hattori-Stong.

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.

Classification : 05A40, 55N22
Keywords: umbral calculus, Hattori-Stong theorems
Mot clés : calcul umbral, théorèmes de Hattori-Stong

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