All polynomials of binomial type are represented by Abel polynomials
An introduction to finite fibonomial calculus
This is an indicatory presentation of main definitions and theorems of Fibonomial Calculus which is a special case of ψ-extented Rota's finite operator calculus [7].
An umbral calculus for polynomials characterizing tensor products.
Compositional inversion of triangular sets of series.
Extended finite operator calculus-an example of algebraization of analysis
“A Calculus of Sequences” started in 1936 by Ward constitutes the general scheme for extensions of classical operator calculus of Rota-Mullin considered by many afterwards and after Ward. Because of the notation we shall call the Ward's calculus of sequences in its afterwards elaborated form-a ψ-calculus. The ψ-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota-Mullin or equivalently-of umbral calculus of Roman and Rota. At the same time this...
Extensions of umbral calculus II: double delta operators, Leibniz extensions and Hattori-Stong theorems
We continue our programme of extending the Roman-Rota umbral calculus to the setting of delta operators over a graded ring with a view to applications in algebraic topology and the theory of formal group laws. We concentrate on the situation where 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...
Formal calculus and umbral calculus.
Fully degenerate poly-Bernoulli numbers and polynomials
In this paper, we introduce the new fully degenerate poly-Bernoulli numbers and polynomials and inverstigate some properties of these polynomials and numbers. From our properties, we derive some identities for the fully degenerate poly-Bernoulli numbers and polynomials.
Identities arising from higher-order Daehee polynomial bases
Here we will derive formulas for expressing any polynomial as linear combinations of two kinds of higherorder Daehee polynomial basis. Then we will apply these formulas to certain polynomials in order to get new and interesting identities involving higher-order Daehee polynomials of the first kind and of the second kind.
Non-Archimedean Umbral Calculus
On a generalized recurrence for Bell numbers.
On an expansion theorem in the finite operator calculus of G.-C. Rota.
On polynomials of Sheffer type arising from a Cauchy problem.
Some congruences concerning the Bell numbers.
Some summation formulae in commutative Leibniz algebras with logarithms
Sur la formule d’inversion de Lagrange
On se propose de démontrer que la formule d’inversion de Lagrange est encore valide sur un anneau commutatif, même pour une série ayant quelques termes à coefficients nilpotents avant le terme de degré 1 (dont le coefficient est inversible). On n’use que de techniques algébriques.
Symmetric identity for polynomial sequences satisfying
Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying with a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, ApostolEuler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.
The identity of three classes of polynomials.
The monotone cumulants
In the present paper we define the notion of generalized cumulants which gives a universal framework for commutative, free, Boolean and especially, monotone probability theories. The uniqueness of generalized cumulants holds for each independence, and hence, generalized cumulants are equal to the usual cumulants in the commutative, free and Boolean cases. The way we define (generalized) cumulants needs neither partition lattices nor generating functions and then will give a new viewpoint to cumulants....
The umbral transfer-matrix method. III: Counting animals.