Free martingale polynomials
- 作者：Anshelevich ; Michael
- 关键词：primary 46L54 ; secondary 05A40 ; 05E35 ; 60G44 ; 47H20 ; Free convolution semigroups ; Free Meixner systems ; Free Sheffer systems ; Free umbral calculus
- 刊名：Journal of Functional Analysis
- 出版年：2003
- 期刊代码：75_00221236
- 类别：mt
- 出版时间：June 20, 2003
- 卷：201
- 期：1
- 页码：228-261
- 文件大小：324 K
摘要
In this paper we investigate the properties of free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. Next, we show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials. Finally, we indicate how Rota's finite operator calculus can be modified for the free context.