The Umbral Calculus

  title={The Umbral Calculus},
  author={Steven Roman},
In this chapter, we give a brief introduction to a relatively new subject, called the umbral calculus. This is an algebraic theory used to study certain types of polynomial functions that play an important role in applied mathematics. We give only a brief introduction to the subject — emphasizing the algebraic aspects rather than the applications. For more on the umbral calculus, we suggest The Umbral Calculus, by Roman [1984]. 
Umbral Calculus And Euler Polynomials
In this paper, some properties of Euler polynomials arising from umbral calculus are studied and some interesting identities of Eucharist polynomsials are given using their results. Expand
General power umbral calculus in several variables
Abstract In this paper we give a version of multivariate general power umbral calculus with the starting point based on the theory given by S. Roman and A. Brini. We define a topological linear spaceExpand
A note on harmonic numbers, umbral calculus and generating functions
Methods of umbral calculus and algebraic nature are used to make some progress in the theory of generating functions involving harmonic numbers and it is shown that the method can be applied to find a common bridge with the Theory of well-known generating functions. Expand
Umbral calculus associated with Frobenius-type Eulerian polynomials
In this paper, we study some properties of several polynomials arising from umbral calculus. In particular, we investigate the properties of orthogonality type of the Frobeniustype EulerianExpand
More on the umbral calculus, with emphasis on the q-umbral calculus
Abstract The interrelationship between distinct umbral calculi is studied. These ideas are applied in particular to q -umbral calculus, which shows how Andrews' q -theory relates to the q -theoryExpand
Baxter Algebras and the Umbral Calculus
  • Li Guo
  • Mathematics, Computer Science
  • Adv. Appl. Math.
  • 2001
A characterization of the umbral calculus in terms of Baxter algebra leads to a natural generalization of the Umbral calculus that include the classical umbrals calculus in a family of $\lambda$-umbral calculi parameterized by $\ lambda$ in the base ring. Expand
Umbral calculus in Ore extensions
Abstract The aim of the paper is to show the existence of some ingredients for an umbral calculus on some Ore extensions, in a manner analogous to Rota's classical umbral calculus which deals with aExpand
The classical umbral calculus
A rigorous presentation of the umbral calculus, as formerly applied heuristically by Blissard, Bell, Riordan, and others is given. As an application, the basic identities for Bernoulli numbers, asExpand
Natural Exponential Families and Umbral Calculus
We use the Umbral Calculus to investigate the relation between natural exponential families and Sheffer polynomials. As a corollary, we obtain a new transparent proof of Feinsilver’s theorem whichExpand
Apostol-Euler polynomials arising from umbral calculus
In this paper, by using the orthogonality type as defined in the umbral calculus, we derive an explicit formula for several well-known polynomials as a linear combination of the Apostol-EulerExpand


Some remarks on Rota's umbral calculus
Summary It is shown that Rota's theory of Sheffer polynomials can be generalized to the quotient field of the ring of formal power series in 1 x . As a special case we give some applications to theExpand
The Lagrange interpolation formula and Stirling numbers
and the formulas may be used to extend the definition of Si(n, k) and S2(n, k) for arbitrary real n. In a previous paper [2] the writer has proved several apparently new formulas relating the twoExpand
Some polynomials defined by generating relations
In an attempt to present a unified treatment of the various polynomial systems introduced from time to time, new generating functions are given for the sets of polynomials {Sn?/ (x; x)} andExpand
Character analogues of the poisson and Euler-MacLaurin summation formulas with applications
Abstract Character versions of the Poisson and Euler-Maclaurin summation formulas are derived. Instead of Bernoulli numbers and polynomials which appear in the classical Euler-Maclaurin formula,Expand
Transformation of certain bilinear generating functions
SummaryErdélyi's generalizations of the Hardy-Hille formula are extended to series involving arbitrary coefficients; see(1.6), …,(1.9) below. These identities may be thought of as identities inExpand
An exposá of the mullin-rota theory of polynomials of binomial type
The basic theorems of the Mullin-Rota theory of polynomials of binomial type are rederived here by a combination of the classical theory of generating functions and Rota's operator theoreticalExpand
Homomorphisms and extensions of modules over certain differential polynomial rings
There is a well-known simple hereditary noethcrian domain L3; this is the @-algebra (@ a field of characteristic zero) with two generators x, y subject to the relation xy -yx = 1. A is isomorphic toExpand
Appell polynomial expansions and biorthogonal expansions in Banach spaces
ABSTRACT. Let {Pkl0 denote the sequence of Appell polynomials generated by an analytic function ?) with the property that the power series for 0 = 1/+ has a larger radius of convergence than theExpand
Polynomial expansions and generating functions
Abstract We study expansions in polynomials {Pn(x)}∞o generated by ∑∞n = o Pn(x)tn = A(t) φ(xtkθ(t)), θ(0) ≠ 0, and ∑∞n = 0 Pn(x)tn = ∑kj = 1 Aj(t) φ(xtϵj), ϵ1,…,ϵk being the k roots of unity. TheExpand
Power series in one variable
Abstract This paper is concerned with the ring A of all complex formal power series and the group G of substitution-invertible formal series. The two main questions of interest will be these. How canExpand