Orthogonal polynomials with a resolvent-type generating function

Anshelevich, Michael

doi:10.1090/S0002-9947-08-04368-7

Orthogonal polynomials with a resolvent-type generating function
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Trans. Amer. Math. Soc. 360 (2008), 4125-4143 Request permission

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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