Set maps, umbral calculus, and the chromatic polynomial

We survey the mathematical literature on umbral calculus (otherwise known as the calculus of finite differences) from its roots in the 19th century (and earlier) as a set of “magic rules” for lowering and raising indices, through its rebirth in the 1970’s as Rota’s school set it on a firm logical foundation using operator methods, to the current state of the art with numerous generalizations and applications. The survey itself is complemented by a fairly complete bibliography (over 500 references) which we expect to update regularly.

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Voigt Transform and Umbral Image

Mathematical and Computational Applications ◽

10.3390/mca25030049 ◽

2020 ◽

Vol 25 (3) ◽

pp. 49

Author(s):

Silvia Licciardi ◽

Rosa Maria Pidatella ◽

Marcello Artioli ◽

Giuseppe Dattoli

Keyword(s):

Spectroscopic Studies ◽

Higher Order ◽

Point Of View ◽

The Other ◽

Pivotal Role ◽

Umbral Calculus ◽

Laguerre Functions ◽

Hermite And Laguerre Functions ◽

Higher Order Functions ◽

Voigt Function

In this paper, we show that the use of methods of an operational nature, such as umbral calculus, allows achieving a double target: on one side, the study of the Voigt function, which plays a pivotal role in spectroscopic studies and in other applications, according to a new point of view, and on the other, the introduction of a Voigt transform and its possible use. Furthermore, by the same method, we point out that the Hermite and Laguerre functions, extension of the corresponding polynomials to negative and/or real indices, can be expressed through a definition in a straightforward and unified fashion. It is illustrated how the techniques that we are going to suggest provide an easy derivation of the relevant properties along with generalizations to higher order functions.

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Umbral Calculus in Non-Archimedean Analysis

P-Adic Functional Analysis ◽

10.1201/9780203908143.ch26 ◽

2001 ◽

Author(s):

Ann Verdoodt

Keyword(s):

Umbral Calculus

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Applications of q-Umbral Calculus to Modied Apostol Type q-Bernoulli Polynomials

10.20944/preprints201712.0118.v1 ◽

2017 ◽

Author(s):

Mehmet Acikgoz ◽

Resul Ates ◽

Ugur Duran ◽

Serkan Araci

Keyword(s):

Linear Combination ◽

Generating Function ◽

Bernoulli Polynomials ◽

Umbral Calculus

This article aims to identify the generating function of modied Apostol type q-Bernoulli polynomials. With the aid of this generating function, some properties of modied Apostol type q-Bernoulli polynomials are given. It is shown that aforementioned polynomials are q-Appell. Hence, we make use of these polynomials to have applications on q-Umbral calculus. From those applications, we derive some theorems in order to get Apostol type modied q-Bernoulli polynomials as a linear combination of some known polynomials which we stated in the paper.

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Link invariants, the chromatic polynomial and the Potts model

Advances in Theoretical and Mathematical Physics ◽

10.4310/atmp.2010.v14.n2.a4 ◽

2010 ◽

Vol 14 (2) ◽

pp. 507-540 ◽

Cited By ~ 6

Author(s):

Paul Fendley ◽

Vyacheslav Krushkal

Keyword(s):

Potts Model ◽

Chromatic Polynomial ◽

Link Invariants

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Algebraic Properties of Chromatic Roots

The Electronic Journal of Combinatorics ◽

10.37236/6578 ◽

2017 ◽

Vol 24 (1) ◽

Cited By ~ 1

Author(s):

Peter J. Cameron ◽

Kerri Morgan

Keyword(s):

Algebraic Integer ◽

Tutte Polynomial ◽

Chromatic Polynomial ◽

Galois Groups ◽

Algebraic Numbers ◽

Isaac Newton ◽

Isaac Newton Institute ◽

Chromatic Root ◽

The Subject ◽

Chromatic Roots

A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

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A Quasisymmetric Function Generalization of the Chromatic Symmetric Function

The Electronic Journal of Combinatorics ◽

10.37236/518 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 6

Author(s):

Brandon Humpert

Keyword(s):

Symmetric Function ◽

Chromatic Polynomial ◽

Quasisymmetric Function ◽

Quasisymmetric Functions ◽

Acyclic Orientations ◽

Chromatic Symmetric Function

The chromatic symmetric function $X_G$ of a graph $G$ was introduced by Stanley. In this paper we introduce a quasisymmetric generalization $X^k_G$ called the $k$-chromatic quasisymmetric function of $G$ and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of $X_G$ to $\chi_G(\lambda)$, the chromatic polynomial, we also define a generalization $\chi^k_G(\lambda)$ and show that evaluations of this polynomial for negative values generalize a theorem of Stanley relating acyclic orientations to the chromatic polynomial.

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Umbral Calculus and the Frobenius-Euler Polynomials

Abstract and Applied Analysis ◽

10.1155/2013/871512 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Dae San Kim ◽

Taekyun Kim ◽

Sang-Hun Lee

Keyword(s):

Umbral Calculus ◽

Euler Polynomials

We study some properties of umbral calculus related to the Appell sequence. From those properties, we derive new and interesting identities of the Frobenius-Euler polynomials.

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