Towards psi-extension of Rota`s Finite Operator Calculus

Maple umbral calculus package

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.

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.

Finite free convolutions via Weingarten calculus

We consider the three finite free convolutions for polynomials studied in a recent paper by Marcus, Spielman, and Srivastava. Each can be described either by direct explicit formulae or in terms of operations on randomly rotated matrices. We present an alternate approach to the equivalence between these descriptions, based on combinatorial Weingarten methods for integration over the unitary and orthogonal groups. A key aspect of our approach is to identify a certain \emph{quadrature property}, which is satisfied by some important series of subgroups of the unitary groups (including the groups …

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…

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.

幂体重新审视——一种古老的符号方法

Poweroids revisited - an old symbolic approach

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。