Maple umbral calculus package
英文摘要：
We are developing a Maple package of functions related to Rota's Umbral Calculus. A Mathematica version of this package is being developed in parallel.
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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)=\sum_{T\uplus 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.
math.CO
The classical umbral calculus: Sheffer sequences
Following the approach of Rota and Taylor \cite{SIAM}, we present an innovative theory of Sheffer sequences in which the main properties are encoded by using umbrae. This syntax allows us noteworthy computational simplifications and conceptual clarifications in many results involving Sheffer sequences. To give an indication of the effectiveness of the theory, we describe applications to the well-known connection constants problem, to Lagrange inversion formula and to solving some recurrence relations.
math.CO
Riordan arrays and applications via the classical umbral calculus
We use the classical umbral calculus to describe Riordan arrays. Here, a Riordan array is generated by a pair of umbrae, and this provides efficient proofs of several basic results of the theory such as the multiplication rule, the recursive properties, the fundamental theorem and the connection with Sheffer sequences. In particular, we show that the fundamental theorem turns out to be a reformulation of the umbral Abel identity. As an application, we give an elementary approach to the problem of extending integer powers of Riordan arrays to complex powers in such a way that additivity of the …
math.CO
Applications of the classical umbral calculus
We describe applications of the classical umbral calculus to bilinear generating functions for polynomial sequences, identities for Bernoulli and related numbers, and Kummer congruences.
math.COmath.NT
Baxter Algebras and Umbral Calculus
We apply recent constructions of free Baxter algebras to the study of the umbral calculus. We give a characterization of the umbral calculus in terms of Baxter algebra. This characterization leads to a natural generalization of the umbral calculus that include the classical umbral calculus in a family of $λ$-umbral calculi parameterized by $λ$ in the base ring.
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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 $\mathbb F_q$-linear functions are constructed.
math.COmath.NT
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运算影子微积分
Operational Umbral Calculus
本文提出了一种基于线性算子的影子微积分理论重建方法，展示了生成函数在证明高级结果中的应用，并讨论了delta算子的伪反演。
math.COmath.NT
New degenerated polynomials arising from non-classical Umbral Calculus
We introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function and concepts of the Umbral Calculus associated with it. Also, we present generalizations of some familiar identities and connection between these kinds of Bernoulli, Euler, and Genocchi polynomials. Moreover, we establish a new analogue of the Euler identity for the degenerate Bernoulli numbers.
math.COmath.NT
Formal calculus and umbral calculus
In this paper we use the viewpoint of the formal calculus underlying vertex operator algebra theory to study certain aspects of the classical umbral calculus and we introduce and study certain operators generalizing the classical umbral shifts. We begin by calculating the exponential generating function of the higher derivatives of a composite function, following a short, elementary proof which naturally arose as a motivating computation related to a certain crucial "associativity" property of an important class of vertex operator algebras. Very similar (somewhat forgotten) proofs had appeared…
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More on the Bernoulli and Taylor Formula for Extended Umbral Calculus
One delivers here the extended Bernoulli and 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. Recent publications do confirm this historically established experience. Its links via umbrality to combinatorics are known at least since Rota and Mullin source papers then up to recently extended by many authors to be indicated in the sequel.
cs.DMmath.CO
向量空间上的阴影演算
Umbral calculus over a vector space
Swansea University（斯旺西大学）
本文在任意向量空间上建立无基阴影演算，定义二项式型与谢弗序列，证明谢弗算子构成同构于里奥丹群的群，并给出代数提升方法。
math.CO
Information on some recent applications of umbral extensions to discrete mathematics
At the first part of the paper we show how specific umbral extensions of the Stirling numbers of the second kind result in new type of Dobinski-like formulas. In the second part among others one recovers how and why Ward solution of uncountable family of extended difference calculus nonhomogeneous equations extends to Ward-Appell polynomials case . Illustrative specifications to q-calculus case and fibonomial calculus case are made explicit due to the usage of the so called upside down notation for objects of extended finite operator calculus .
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关于影算子的公式和分数指数
On formulas and fractional exponents for umbral operators
本文提出新的影算子公式，揭示其与迭代理论的联系，定义了影算子的分数次幂，并综合了操作微积分中的多项成果，扩展了拉格朗日多项式。
math.CAmath.CO
基于Rota的伞微积分方法的Bell数B型类似物
Type-B analogue of Bell numbers using Rota's Umbral calculus approach
该研究将Rota、Tanny基于伞微积分得到的Bell数相关结果，推广到Coxeter B型集合划分框架，得到Bell数的B型类似物。
cs.DMmath.CO
$ψ$-Appell polynomials` solutions of an umbral difference nonhomogeneous equation
One discovers why the solution of generalized umbral calculus difference nonhomogeneous equation in the form recently proposed by the author extends here now to generalized appellian delta operator and corresponding polynomials case almost automatically. The reason for that is just the proper framework of the generalized finite operator calculus recently being developed by the present author.
cs.DMmath.CO
Towards psi-extension of Rota`s Finite Operator Calculus
A class of extended umbral calculi in operator form is presented. Extensions of all basic theorems of classical Finite Operator Calculus are shown to hold. The impossibility of straightforward extending of quantum q-plane formulation of the q-umbral caculus to the general psi-calculus case is demonstrated.
cs.DMmath.CO
海森堡群上协变变换的嬗变与扩展伞形原理
Transmutations from the Covariant Transform on the Heisenberg Group and an Extended Umbral Principle
本文以海森堡群Weyl代数的升降算符表示为统一框架，研究伞形演算、广义平移与嬗变及算符符号演算之间的联系，回顾经典结果并揭示新机遇。
math.APmath.COmath.FA
Umbral nature of the Poisson random variables
Extending the rigorous presentation of the classical umbral calculus given by Rota and Taylor in 1994, the so-called partition polynomials are interpreted with the aim to point out the umbral nature of the Poisson random variables. Among the new umbrae introduced, the main tool is the partition umbra that leads also to a simple expression of the functional composition of the exponential power series. Moreover a new short proof of the Lagrange inversion formula is given.
math.COmath.PR
On Simple Characterisations of Sheffer psi- polynomials and Related Propositions of the Calculus of Sequences
A calculus of sequences started in 1936 opened the way for future extensions of umbral calculus in its finite operator form. Because of historically established notation we call it the psi-calculus.It appears in parts to be almost automatic extension of the standard classical finite operator calculus.
cs.DMmath.CO
Operational Methods in the Study of Sobolev-Jacobi Polynomials
Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-called umbral image technique. Besides providing a class of new formulae for generalized hypergeometric functions and an implementation of series manipulations for computing lacunary generating functions, our main application of these techniques is the study of Sobolev-Jacobi polynomials. Motivated by applications to theoretical chemistry, we moreover…
math-phmath.CAmath.CO
Weingarten Calculus and the IntHaar Package for Integrals over Compact Matrix Groups
In this paper, we present a uniform formula for the integration of polynomials over the unitary, orthogonal, and symplectic groups using Weingarten calculus. From this description, we further simplify the integration formulas and give several optimizations for various cases. We implemented the optimized Weingarten calculus in Maple in a package called IntHaar for all three compact groups. Here we will discuss its functions, provide instructions for the package, and produce some examples of computed integrals.
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Polynomial Sequences of Binomial Type and Path Integrals
Polynomial sequences $p_n(x)$ of binomial type are a principal tool in the umbral calculus of enumerative combinatorics. We express $p_n(x)$ as a \emph{path integral} in the ``phase space'' $\Space{N}{} \times {[-π,π]}$. The Hamiltonian is $h(ϕ)=\sum_{n=0}^\infty p_n'(0)/n! e^{inϕ}$ and it produces a Schrödinger type equation for $p_n(x)$. This establishes a bridge between enumerative combinatorics and quantum field theory. It also provides an algorithm for parallel quantum computations. Keywords: Feynman path integral, umbral calculus, polynomial sequence of binomial type, token, Schrödinger …
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An example of algebraization of analysis and Fibonacci cobweb poset characterization
In recent Kwasniewski's papers inspired by O. V. Viskov it was shown that 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. The first part of the article is the review of the recent author's contribution. The main definitions and theorems of Finite Fibonomial Operator Calculus which is a special case of $ψ$-extented Rota's finite operat…
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A symbolic method for k-statistics
Trough the classical umbral calculus, we provide new, compact and easy to handle expressions of k-statistics, and more in general of U-statistics. In addition such a symbolic method can be naturally extended to multivariate case and to generalized k-statistics.
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幂体重新审视——一种古老的符号方法
Poweroids revisited - an old symbolic approach
The University of Manchester（曼彻斯特大学）
本文重审Jeffery 1861年的有限差分计算，扩展至一般delta算子，用于简洁证明多项式二项类型理论中的定理，提供紧凑的符号复合、二项性质和连接常数处理方法。
cs.NAmath-phmath.CA
$q$-导数语法学
$q$-Derivative Grammar
Université de Strasbourg et CNRS（斯特拉斯堡大学与法国国家科学研究中心）；Tianjin University（天津大学）；Harbin Institute of Technology（哈尔滨工业大学）
本文提出$q$-导数语法学，建立基本框架并发展计算$q$-指数生成函数的算术，用于研究$q$-欧拉、$q$-罗斯勒和$q$-安德烈多项式及其生成函数和递推关系。
math.CO
A unifying framework for $k$-statistics, polykays and their multivariate generalizations
Through the classical umbral calculus, we provide a unifying syntax for single and multivariate $k$-statistics, polykays and multivariate polykays. From a combinatorial point of view, we revisit the theory as exposed by Stuart and Ord, taking into account the Doubilet approach to symmetric functions. Moreover, by using exponential polynomials rather than set partitions, we provide a new formula for $k$-statistics that results in a very fast algorithm to generate such estimators.
math.COmath.STstat.TH
