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Extensions of umbral calculus: Penumbral coalgebras and generalized Bernoulli numbers. (English) Zbl 0631.05002
The aim of this long paper is to extend the formalism of the umbral calculus, as developed by S. Roman [The umbral calculus (1984; Zbl 0536.33001)] and G.-C. Rota, to the setting of graded rings such as one commonly meets in algebraic topology. This paper serves as the foundation for a number of subsequent studies by the author, devoted to further developments of umbral calculus and the exploration of connections with algebraic topology and combinatorics.
The central concept is the notion of a \(\Delta\)-operator on the binomial coalgebra \(E_*[x]\) over a graded ring \(E_*\), consisting of the polynomials over \(E_*\) with coproduct sending \(x^ n\) to \(\sum^{n}_{k=0}\left( \begin{matrix} n\\ k\end{matrix} \right)x^ k\otimes x^{n-k}\). The prototypical \(\Delta\)-operator is the differentiation operator D, satisfying \(Dx^ n=nx^{n-1}\). The general \(\Delta\)- operator is determined by a sequence of elements \(\phi =(1,\phi_ 1,\phi_ 2,...)\) of \(E_*\) with \(\phi_ k\in E_{2k}\), and in standard umbral notation is defined by \[ \Delta x^ n=(x+\phi)^ n- x^ n,\quad \phi^ k\equiv \phi_{k-1}. \] The dual algebra \(E^*((D))\) to the coalgebra \(E_*[x]\) is called the umbral algebra; it is a divided power algebra which one customarily identifies with an algebra of operators on \(E_*[x].\)
The notions of E-coalgebra and E-algebra are then introduced, where \(E=(E_*,\Delta)\) denotes a graded ring \(E_*\) together with a \(\Delta\)-operator on \(E_*[x]\). Particular examples, denoted A(E) and E[[\(\Delta\) ]], are constructed and shown to be universal. The penumbral coalgebra \(\Pi\) (E) of the title is introduced in order to study the torsion free part of A(E). Not every \(\Delta\)-operator satisfies a product rule of the form \[ \Delta (p(x)q(x))=p(x)\Delta q(x)+q(x)\Delta p(x)+\sum_{i,j}e_{ij}(\Delta^ ip(x))(\Delta^ jq(x)). \] Those that do are called Leibniz \(\Delta\)-operators, and are closely related to formal group theory.
As a final topic, the well-known theorems of von Staudt concerning the classical Bernoulli numbers are extended to sequences of generalized Bernoulli numbers which arise naturally in the present setting.
Reviewer: P.Landweber

MSC:
05A40 Umbral calculus
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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