Extensions of umbral calculus: Penumbral coalgebras and generalized Bernoulli numbers.

*(English)*Zbl 0631.05002The 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.

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 |

##### Keywords:

umbral calculus; graded rings; \(\Delta \)-operator; E-coalgebra; E- algebra; penumbral coalgebra; generalized Bernoulli numbers
Full Text:
DOI

**OpenURL**

##### References:

[1] | Adams, J.F, On the groups J(X), II, Topology, 3, 137-171, (1965) · Zbl 0137.16801 |

[2] | Adams, J.F, Stable homotopy and generalised homology, (1972), Chicago Univ. Press Chicago · Zbl 0309.55016 |

[3] | Baker, A, Combinatorial and arithmetic identities based on formal group laws, (1984), Manchester Univ.,, preprint |

[4] | Borevich, Z.I; Shafarevich, I.R, Number theory, (1966), Academic Press New York · Zbl 0145.04902 |

[5] | Dibag, I, An analogue of the von-staudt clausen theorem, J. algebra, 87, 332-341, (1984) · Zbl 0536.10012 |

[6] | Hazewinkel, M, Formal groups and applications, (1978), Academic Press New York · Zbl 0454.14020 |

[7] | Joni, S.A; Rota, G.-C, Coalgebras and bialgebras in combinatorics, Stud. appl. math., 61, 93-139, (1979) · Zbl 0471.05020 |

[8] | MacLane, S, Homology, (1963), Springer-Verlag Berlin · Zbl 0133.26502 |

[9] | MacLane, S; Birkhoff, G, Algebra, (1967), Macmillan, Co., New York · Zbl 0153.32401 |

[10] | Miller, H.R, Universal Bernoulli numbers and the S1-transfer, (), 437-449 |

[11] | Mullin, R; Rota, G.-C, On the foundations of combinatorial theory. III. theory of binomial enumeration, (), 168-213 |

[12] | Riordan, J, An introduction to combinatorial analysis, (1980), Princeton Univ. Press Princeton, N.J · Zbl 0436.05001 |

[13] | Roman, S, The umbral calculus, (1984), Academic Press New York · Zbl 0536.33001 |

[14] | Roman, S; Rota, G.-C, The umbral calculus, Advan. in math., 27, 95-188, (1978) · Zbl 0375.05007 |

[15] | Schwartz, L, Operations d’Adams en K-homolgie et applications, Bull. soc. math. France, 109, 237-257, (1981) · Zbl 0472.55013 |

[16] | Yang, K.-W, Integration in the umbral calculus, J. math. anal. appl., 74, 200-211, (1980) · Zbl 0453.05009 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.