632 research outputs found
Umbral Methods and Harmonic Numbers
The theory of harmonic based function is discussed here within the framework
of umbral operational methods. We derive a number of results based on
elementary notions relying on the properties of Gaussian integrals.Comment: 6 page
Umbral methods and harmonic numbers
The theory of harmonic-based functions is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals
umbral methods combinatorial identities and harmonic numbers
We analyse and demonstrate how umbral methods can be applied for the study of the problems, involving combinatorial calculus and harmonic numbers. We demonstrate their efficiency and we find the general procedure to frame new and existent identities within a unified framework, amenable of further generalizations
Inverse derivative operator and umbral methods for the harmonic numbers and telescopic series study
The formalism of differ-integral calculus, initially developed to treat differential operators of fractional order, realizes a complete symmetry between differential and integral operators. This possibility has opened new and interesting scenarios, once extended to positive and negative order derivatives. The associated rules offer an elegant, yet powerful, tool to deal with integral operators, viewed as derivatives of order-1. Although it is well known that the integration is the inverse of the derivative operation, the aforementioned rules offer a new mean to obtain either an explicit iteration of the integration by parts or a general formula to obtain the primitive of any infinitely differentiable function. We show that the method provides an unexpected link with generalized telescoping series, yields new useful tools for the relevant treatment, and allows a practically unexhausted tool to derive identities involving harmonic numbers and the associated generalized forms. It is eventually shown that embedding the differ-integral point of view with techniques of umbral algebraic nature offers a new insight into, and the possibility of, establishing a new and more powerful formalism
Multivariate time-space harmonic polynomials: a symbolic approach
By means of a symbolic method, in this paper we introduce a new family of
multivariate polynomials such that multivariate L\'evy processes can be dealt
with as they were martingales. In the univariate case, this family of
polynomials is known as time-space harmonic polynomials. Then, simple
closed-form expressions of some multivariate classical families of polynomials
are given. The main advantage of this symbolic representation is the plainness
of the setting which reduces to few fundamental statements but also of its
implementation in any symbolic software. The role played by cumulants is
emphasized within the generalized Hermite polynomials. The new class of
multivariate L\'evy-Sheffer systems is introduced.Comment: In pres
Multivariate Bernoulli and Euler polynomials via L\'evy processes
By a symbolic method, we introduce multivariate Bernoulli and Euler
polynomials as powers of polynomials whose coefficients involve multivariate
L\'evy processes. Many properties of these polynomials are stated
straightforwardly thanks to this representation, which could be easily
implemented in any symbolic manipulation system. A very simple relation between
these two families of multivariate polynomials is provided
Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice
`Umbral calculus' deals with representations of the canonical commutation
relations. We present a short exposition of it and discuss how this calculus
can be used to discretize continuum models and to construct representations of
Lie algebras on a lattice. Related ideas appeared in recent publications and we
show that the examples treated there are special cases of umbral calculus. This
observation then suggests various generalizations of these examples. A special
umbral representation of the canonical commutation relations given in terms of
the position and momentum operator on a lattice is investigated in detail.Comment: 19 pages, Late
- …