Baxter Algebras and Umbral Calculus

Baxter Algebras and Differential Algebras

A Baxter algebra is a commutative algebra $A$ that carries a generalized integral operator. In the first part of this paper we review past work of Baxter, Miller, Rota and Cartier in this area and explain more recent work on explicit constructions of free Baxter algebras that extended the constructions of Rota and Cartier. In the second part of the paper we will use these explicit constructions to relate Baxter algebras to Hopf algebras and give applications of Baxter algebras to the umbral calculus in combinatorics.

Rota-Baxter Algebra. The Combinatorial Structure of Integral Calculus

Gian-Carlo Rota suggested in one of his last articles the problem of developing a theory around the notion of integration algebras, complementary to the already existing theory of differential algebras. This idea was mainly motivated by Rota's deep appreciation for Kuo-Tsai Chen's seminal work on iterated integrals. As a starting point for such a theory of integration algebras Rota proposed to consider a particular operator identity first introduced by the mathematician Glen Baxter. Later it was coined Rota-Baxter identity. In this article we briefly recall basic properties of Rota--Baxter alg…

Monads and distributive laws for Rota-Baxter and differential algebras

In recent years, algebraic studies of the differential calculus and integral calculus in the forms of differential algebra and Rota-Baxter algebra have been merged together to reflect the close relationship between the two calculi through the First Fundamental Theorem of Calculus. In this paper we study this relationship from a categorical point of view in the context of distributive laws which can be tracked back to the distributive law of multiplication over addition. The monad giving Rota-Baxter algebras and the comonad giving differential algebras are constructed. Then a mixed distributive…

Hypergeometric Functions and Carlitz Differential Equations over Function Fields

The paper is a survey of recent results in analysis of additive functions over function fields motivated by applications to various classes of special functions including Thakur's hypergeometric function. We consider basic notions and results of calculus, analytic theory of differential equations with Carlitz derivatives (including a counterpart of regular singularity), umbral calculus, holonomic modules over the Weyl-Carlitz ring.

从迭代积分和时序演算到Hopf与Rota-Baxter代数

From iterated integrals and chronological calculus to Hopf and Rota-Baxter algebras

本综述统一呈现迭代积分与时序演算通向Hopf与Rota-Baxter代数的三条发展路线，提出几何与代数两种视角，涵盖非交换Rota-Baxter代数及相关结构。

On Differential Rota-Baxter Algebras

A Rota-Baxter operator of weight $λ$ is an abstraction of both the integral operator (when $λ=0$) and the summation operator (when $λ=1$). We similarly define a differential operator of weight $λ$ that includes both the differential operator (when $λ=0$) and the difference operator (when $λ=1$). We further consider an algebraic structure with both a differential operator of weight $λ$ and a Rota-Baxter operator of weight $λ$ that are related in the same way that the differential operator and the integral operator are related by the First Fundamental Theorem of Calculus. We construct free objec…