Orthogonal polynomials with a resolvent-type generating function
| dc.creator | Anshelevich, Michael | |
| dc.date | 2004-10-22 | |
| dc.date | 2006-07-03 | |
| dc.date.accessioned | 2026-07-07T09:49:54Z | |
| dc.date.available | 2026-07-07T09:49:54Z | |
| dc.description | 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. | |
| dc.description | 19 pages; minor improvements | |
| dc.identifier | https://arxiv.org/abs/math/0410482 | |
| dc.identifier | http://arxiv.org/abs/math/0410482 | |
| dc.identifier | Trans. Amer. Math. Soc. 360 (2008), 4125-4143 | |
| dc.identifier | doi:10.1090/S0002-9947-08-04368-7 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/164761 | |
| dc.subject | Combinatorics | |
| dc.subject | Operator Algebras | |
| dc.subject | Primary 05E35; Secondary 46L54, 33C47 | |
| dc.title | Orthogonal polynomials with a resolvent-type generating function | |
| dc.type | text |