Umbral Calculus Research Articles

Using Rota's Umbral calculus to enumerate Stanley's P-partitions

Gian-Carlo's powerfull Umbral calculus, suitably automated, is used to enumerate large families of Richard Stanley's P-partitions.

BASIC APPELL SEQUENCES

We study a subclass of Sheffer sequences denominated Appell sequences. Using methods of the so-called umbral calculus in conjunction with general results of nuclear Fréchet spaces and the theory of Köthe sequence spaces, we give necessary and sufficient conditions for an Appell sequence to be a basis.

Set maps, umbral calculus, and the chromatic polynomial

Some important properties of the chromatic polynomial also hold for any polynomial set map satisfying p S ( x + y ) = ∑ T ⊎ U = S p T ( x ) p U ( y ) . Using umbral calculus, we give a formula for the expansion of such a set map in terms of any polynomial sequence of binomial type. This leads to some new expansions of the chromatic polynomial. We also describe a set map generalization of Abel polynomials.

A note on harmonic numbers, umbral calculus and generating functions

We use methods of umbral calculus and algebraic nature to make some progress in the theory of generating functions involving harmonic numbers. We derive old and new results on generating functions in a fairly simple way and show that the method can be applied to find a common bridge with the theory of well-known (rather classical) generating functions.

Monomiality and multi-index multi-variable special polynomials

In this paper, we will use the operational recipes from monomiality and umbral calculus to deal with multi-dimensional polynomials.

Quasi-holonomic modules in positive characteristic

We study modules over the Carlitz ring, a counterpart of the Weyl algebra in analysis over local fields of positive characteristic. It is shown that some basic objects of function field arithmetic, like the Carlitz module, Thakur's hypergeometric polynomials, and analogs of binomial coefficients arising in the function field version of umbral calculus, generate quasi-holonomic modules. This class of modules is, in many respects, similar to the class of holonomic modules in the characteristic zero theory.

Compactly Supported Wavelets Through the Classical Umbral Calculus

We explore compactly supported scaling functions of wavelet theory by means of classical umbral calculus as reformulated by Rota and Taylor. We set a theory of orthonormal scaling umbra which leads to a very simple and elementary proof of Lawton's theorem for umbrae. When umbrae come from a wavelet setting, we recover the usual Lawton condition for the orthonormality of the integer translates of a scaling function.

More on the Bernoulli*–Taylor Formula for Extended Umbral Calculus

One presents here the *ψ-Bernoulli-Taylor* formula of a new sort with the rest term of the Cauchy type recently derived by the author in the case of the so called ψ-difference calculus which constitutes the representative for the purpose case of extended umbral calculus. The central importance of such a type formulas is beyond any doubt - and recent publications do confirm this historically established experience.

A symbolic method for [formula omitted]-statistics

Through the classical umbral calculus, we provide new, compact and easy to handle expressions for k -statistics, and more generally for U -statistics. In addition, this symbolic method can be naturally extended to multivariate cases and to generalized k -statistics.

Representations of monomiality principle with Sheffer-type polynomials and boson normal ordering

We construct explicit representations of the Heisenberg–Weyl algebra [ P , M ] = 1 in terms of ladder operators acting in the space of Sheffer-type polynomials. Thus we establish a link between the monomiality principle and the umbral calculus. We use certain operator identities which allow one to evaluate explicitly special boson matrix elements between the coherent states. This yields a general demonstration of boson normal ordering of operator functions linear in either creation or annihilation operators. We indicate possible applications of these methods in other fields.

Cayley continuants

In 1858 Cayley considered a particular kind of tridiagonal determinants (or continuants). By a direct inspection of the first cases, he conjectured an identity expressing these determinants in terms of certain other determinants considered by Sylvester in 1854. Then Cayley proved the conjectured identity by induction but, as he wrote, he felt unsatisfied with his proof. The main aim of this paper is to give a straightforward proof of Cayley's identity using the method of formal series. Moreover we use this method and umbral calculus techniques to obtain several other identities. Cayley continuants appear in several contexts and in particular in enumerative combinatorics. Mittag–Leffler polynomials, Meixner polynomials of the first kind, the falling and the raising factorials are just few instances of these continuants. They can be interpreted in terms of weighted permutations. Moreover, as we prove in this paper, they also appear in the context of Hankel determinants generated by certain Catalan-like numbers.

An umbral setting for cumulants and factorial moments

We provide an algebraic setting for cumulants and factorial moments via the classical umbral calculus. Our main tools are the compositional inverse of the unity umbra, this being related to logarithmic power series, and a new umbra here introduced, the singleton umbra. We develop formulae that express cumulants, factorial moments and central moments as umbral functions. Nous etudions les cumulants et les moments factoriels par le calcul ombral classique. Les outils principaux sont l’inverse compositionnel de l’ombre unité, lié à la série formelle logarithmique, et un nouvel ombre, ici présenté, l’ombre singleton. De diverses formules sont données exprimant les cumulants, les moments factoriels et les moments centraux par des fonctions ombral.

Umbral calculus in positive characteristic

An umbral calculus over local fields of positive characteristic is developed on the basis of a relation of binomial type satisfied by the Carlitz polynomials. Orthonormal bases in the space of continuous F q -linear functions are constructed.

Umbral calculus, difference equations and the discrete Schrödinger equation

In this paper, we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schrödinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space–time. In this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case. Moreover, systems that are integrable, superintegrable or exactly solvable preserve these properties in the discrete case.

Extended finite operator calculus—an example of algebraization of analysis

Abstract “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 calculus is an example of the algebraization of the analysis—here restricted to the algebra of polynomials. Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus where Q stands for a generalized difference operator, i.e. the one lowering the degree of any polynomial by one. This is a review article based on the recent first author contributions [1]. As the survey article it is supplemented by the short indicatory glossaries of notation and terms used by Ward [2], Viskov [7, 8], Markowsky [12], Roman [28–32] on one side and the Rota-oriented notation on the other side [9–11, 1, 3, 4, 35] (see also [33]).

Counting lattice paths taking steps in infinitely many directions under special access restrictions

We count lattice paths that are confined to the first quadrant by the nature of their step vectors. If no further restrictions apply, a path can go from any point to infinitely many others, but each point on the path has only finitely many predecessors. By “ further restrictions” we mean a boundary line above which the paths may have to stay. Access privilege to the boundary line itself is granted from certain lattice points in the form of a special access step set, which itself may be infinite. We also count the number of paths that contact the weak boundary a given number of times. We approach explicit solutions of such enumeration problems via Sheffer polynomials and functionals, using results of the Umbral Calculus.

Lorentz and Galilei invariance on lattices

We show that the algebraic aspects of Lie symmetries and generalized symmetries in nonrelativistic and relativistic quantum mechanics can be preserved in linear lattice theories. The mathematical tool for symmetry preserving discretizations on regular lattices is the umbral calculus.

Counting lattice paths taking steps in infinitely many directions under special access restrictions

We count lattice paths that are confined to the first quadrant by the nature of their step vectors. If no further restrictions apply, a path can go from any point to infinitely many others, but each point on the path has only finitely many predecessors. By “ further restrictions” we mean a boundary line above which the paths may have to stay. Access privilege to the boundary line itself is granted from certain lattice points in the form of a special access step set, which itself may be infinite. We also count the number of paths that contact the weak boundary a given number of times. We approach explicit solutions of such enumeration problems via Sheffer polynomials and functionals, using results of the Umbral Calculus.

Discreteq-derivatives and symmetries ofq-difference equations

In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many of the properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This q-umbral calculus can be used to provide solutions to linear q-difference equations and q-differential delay equations. To illustrate the method, we will apply the obtained results to the construction of symmetry solutions for the q-heat equation and to solve a linear ordinary second order q-difference equation

Umbral Calculus, Martingales, and Associated Polynomials

In this paper we treat with a relationship between normal martingales [Emery, M. On the Azéma Martingales; Sem. Proba. XXIII, Lec. Notes in Maths 1372; Springer, 1989; 66–87] and some special polynomial families. This problem was studied by Privault et al. [Privault, N.; Sole, J.L.; Vives, J. Chaotic Kabanov Formula for the Azéma Martingales; The Bernoulli Society for Mathematical Statistics and Probability, 2000; Vol. 6, No. 4], they showed under some assumptions which we give later, that Wiener and compensated Poisson processes are the only normal martingales which has an associated family of polynomials according to the Definition 3 in Privault et al. [Privault, N.; Sole, J.L.; Vives, J. Chaotic Kabanov Formula for the Azéma Martingales; The Bernoulli Society for Mathematical Statistics and Probability, 2000; Vol. 6, No. 4]. We study this problem without the authors assumptions in considering all normal martingales, we show more precisely that Wiener process is characterized by Wick polynomials and both Wiener and compensated Poisson processes are characterized by cross polynomials (said also of convolution type). We use here the Umbral Calculus theory [Roman, S.M. The theory of umbral calculus I. J. Math. Anal. Appl. 1982, 87, 58–115] as a new and a fast tool to solve a martingale problem.