Recent contributions to the calculus of finite differences: a survey

线性扩展与牛顿插值的微积分

Calculus of linear extensions and Newton interpolation

本文利用牛顿均差计算Greene和（由偏序集线性扩展确定的有理函数），并推导出牛顿均差与Arnold微分形式的关系，同时得到了由偏序集序列索引的牛顿插值级数的推广。

Extended finite operator calculus as an example of algebraization of analysis

A wardian calculus of sequences started almost seventy years ago constitutes the general scheme for extensions of the classical umbral operator calculus considered by many afterwards . At the same time this calculus is an example of the algebraization of the analysis here restricted to the algebra of formal series. This is a review article based on the recent first author contributions. As the survey article it is supplemented by the short indicatory glossaries of notation and terms used by prominent contributors to the domain.

$ψ$-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.

Free Calculus

I give a survey about my work on combinatorial and probabilistic aspects of free probability theory. In particular, I present the combinatorial description of freeness in terms of free cumulants and I give some ideas of the main results of my joint work with Philippe Biane on a stochastic integration theory for free Brownian motion.

On $ψ$-basic bernoulli-wardian polynomials

The wardian solution of any $ψ$-difference linear nonhomogeneous equation is found in the framework of the generalized finite operator calculus . Specifications to $q$-calculus case and the new one fibonomial calculus case are made explicit.

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.

Orthogonal divided differences and Schubert polynomials, P-functions, and vertex operators

We give several new formulas which are useful for Schubert Calculus associated with the orthogonal groups and related orthogonal degeneracy loci.

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 .