Orthogonal polynomials with a resolvent-type generating function
Orthogonal polynomials with a resolvent-type generating function
复制标题
具有解析型生成函数的正交多项式
DOI：
10.1090/s0002-9947-08-04368-7
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
M. Anshelevich
中科院分区：
文献类型：
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作者：
M. Anshelevich
本文的主题是多个非交换变量的多项式。对于这种与状态正交的多项式，我们证明了 Favard 型递归关系。另一方面，自由谢弗多项式是具有解析型生成函数的非交换变量的多项式族。在这些族中，我们描述了正交的族。它们的递归关系有更特殊的形式；描述它们的最好方法是用正交态的自由累积生成函数来表示，它满足一类二阶差分方程。如果差分方程实际上是一阶的，并且状态是跟踪的，我们表明该状态必然是自由乘积状态的旋转。我们还描述了具有正交自由谢弗多项式的非踪迹无限可分状态的有趣例子。
The subject of this paper are polynomials in multiple non-commuting variables. For polynomials of this type orthogonal with respect to a state, we prove a Favard-type recursion relation. On the other hand, free Sheffer polynomials are a polynomial family in non-commuting variables with a resolvent-type generating function. Among such families, we describe the ones that are orthogonal. Their recursion relations have a more special form; the best way to describe them is in terms of the free cumulant generating function of the state of orthogonality, which turns out to satisfy a type of second-order difference equation. If the difference equation is in fact first order, and the state is tracial, we show that the state is necessarily a rotation of a free product state. We also describe interesting examples of non-tracial infinitely divisible states with orthogonal free Sheffer polynomials.