# Heisenberg algebra, umbral calculus and orthogonal polynomials

@article{Dattoli2008HeisenbergAU, title={Heisenberg algebra, umbral calculus and orthogonal polynomials}, author={Giuseppe Dattoli and Decio Levi and Pavel Winternitz}, journal={Journal of Mathematical Physics}, year={2008}, volume={49}, pages={053509} }

Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation [P,M]=1. In ordinary quantum mechanics, P is the derivative and M the coordinate operator. Here, we shall realize P as a second order differential operator and M as a first order integral one. We show that this makes it possible to solve large classes of differential and integrodifferential equations and to introduce new classes of orthogonal polynomials, related to Laguerre polynomials. These polynomials… Expand

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#### 17 Citations

SOME IDENTITIES OF THE APOSTOL TYPE POLYNOMIALS ARISING FROM UMBRAL CALCULUS

- 2017

Abstract The aim of this paper is to introduce and investigate several new identities related to the uni ed families of Apostol type polynomials. The results presented here are based upon the theory… Expand

(DISCRETE) ALMANSI TYPE DECOMPOSITIONS: AN UMBRAL CALCULUS FRAMEWORK BASED ON osp(1|2) SYMMETRIES

- Mathematics, Physics
- 2011

We introduce the umbral calculus formalism for hypercomplex variables starting from the fact that the algebra of multi- variate polynomials IRŒxshall be described in terms of the generators of the… Expand

Operational, umbral methods, Borel transform and negative derivative operator techniques

- Mathematics
- 2020

ABSTRACT Differintegral methods, namely those techniques using differential and integral operators on the same footing, currently exploited in calculus, provide a fairly unexhausted source of tools… Expand

Operational vs. Umbral Methods and Borel Transform

- Mathematics
- 2019

Differintegral methods, currently exploited in calculus, provide a fairly unexhausted source of tools to be applied to a wide class of problems involving the theory of special functions and not only.… Expand

On Poly-Bernoulli polynomials of the second kind with umbral calculus viewpoint

- Mathematics
- 2015

Poly-Bernoulli polynomials of the second kind were introduced in Kim et al. (Adv. Differ. Equ. 2014:219, 2014) as a generalization of the Bernoulli polynomial of the second kind. Here we investigate… Expand

Barnes-type Peters polynomial with umbral calculus viewpoint

- Mathematics
- 2014

In this paper, we consider the Barnes-type Peters polynomials. We present several explicit formulas and recurrence relations for these polynomials. Also, we establish a connection between our… Expand

Barnes-type Daehee of the first kind and poly-Cauchy of the first kind mixed-type polynomials

- Mathematics
- 2014

In this paper, by considering Barnes-type Daehee polynomials of the first kind as well as poly-Cauchy polynomials of the first kind, we define and investigate the mixed-type polynomials of these… Expand

Integrals of Special Functions and Umbral Methods

- Mathematics
- 2021

We derive integrals of combination of Gauss and Bessel functions, by the use of umbral techniques. We show that the method allows the possibility of pursuing new and apparently fruitful avenues in… Expand

Degenerate poly-Cauchy polynomials with a q parameter

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- 2015

In this paper, the degenerate poly-Cauchy polynomials with a q parameter of the first and the second kind are introduced and their properties are studied. For these polynomials, some explicit… Expand

Degenerate poly-Cauchy polynomials

- Mathematics, Computer Science
- Appl. Math. Comput.
- 2015

Several explicit formulas and recurrence relations for the degenerate poly-Cauchy polynomials are presented and a connection is established between this paper and several known families of polynmials. Expand

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