Orthogonal polynomials with a resolvent-type generating function Academic Article uri icon


  • 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 secondorder 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. 2008 American Mathematical Society.

published proceedings


author list (cited authors)

  • Anshelevich, M.

citation count

  • 20

complete list of authors

  • Anshelevich, Michael

publication date

  • February 2008