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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Orthogonal polynomials with a resolvent-type generating function

Author: Michael Anshelevich
Journal: Trans. Amer. Math. Soc. 360 (2008), 4125-4143
MSC (2000): Primary 05E35; Secondary 46L54, 33C47
Published electronically: February 27, 2008
MathSciNet review: 2395166
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Abstract: 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.

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Additional Information

Michael Anshelevich
Affiliation: Department of Mathematics, University of California, Riverside, California 92521-0135
Address at time of publication: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Received by editor(s): October 25, 2004
Received by editor(s) in revised form: June 7, 2006
Published electronically: February 27, 2008
Additional Notes: This work was supported in part by an NSF grant DMS-0400860
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.