Generating Functions of Cauchy–stieltjes Type for Orthogonal Polynomials

Abstract

We give a free probabilistic interpretation of the multiplicative renormalization method applied with h(x) = (1 − x) −1 . As a byproduct, we give a short proof of the Asai-Kubo-Kuo result on the characterization of the family of probability measures with a Cauchy-Stieltjes type generating functions for orthogonal polynomials. This family turns to be the free Meixner family. We also give representations for the Voiculescu transform of all free Meixner distributions (even in the non-freely infinitely divisible case).

Topics on Meixner families

“…Then we do the same for both Kubo's and Morris's characterizations. In the free setting, a connection is found between Bryc and Ismail's characterization using Cauchy-Stieljes families of quadratic variances and the one given by the authors via Cauchy-Stieltjes-type generating functions of OP ( [10]).…”

“…On the one hand, a combination of [4] and [9] allowed us to characterize the free Meixner distributions and Asai-Kuo-Kubo criterion ( [2]) was used to derive analogous identities to (5) and (6) involving the free cumulant generating function ( [10]). More precisely, let µ be a standard probability distribution with all moments finite.…”

“…where H is analytic around z = 0 such that H(0) = 0, H (0) = 1, X is a random variable in some probability space (Ω, F , P) with law µ = P • X −1 and (P n ) n≥0 is the set of orthogonal polynomials (OP) with respect to µ. ψ is an OP generating After the birth of free probability theory, for which miscellaneous analogs of known results in classical probability theory were derived, the free Meixner family was defined in [4] and covers six probability distributions called analogously to their classical counterparts free Gaussian or Wigner, free Poisson or Marchenko-Pastur, free Gamma, free negative binomial, free Meixner and free binomial, also known as the free stationary Jacobi law ( [13], [15], [16]). Their study produced free versions of some of the above characterizations: the exponential-type OP generating functions are replaced by generating functions of Cauchy-Stieltjes type ( [4], [10], [19]), free Meixner distributions solve a quadratic regression problem in a free setting ( [9]) and have associated Cauchy-Stieltjes type families with linear means and quadratic variances ( [11]). To our best knowledge, no free analogs of Sheffer's and Al-Salam and Chihara's characterizations exist in literature.…”

Contributions to High–Dimensional Analysis under Kolmogorov Condition

“…The Stieltjes transform defined in this section, up to a sign, is often referred to as the Cauchy transform (i.e., Cima et al, 2006;Hiai and Petz, 2000;Nica and Speicher, 2006) or the Stieltjes-Cauchy transform (i.e., Hasebe, 2012;Bożejko and Demni, 2009). In this thesis we use the term Stieltjes transform following the work by Couillet and Debbah (2011).…”

An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions

“…Результаты, рассмотренные выше, имеют некоммутативные аналоги в свободной теории вероятностей [16], [17] (см. также [4], [5], [10], [12], [13], [43] и приведенные там ссылки). По поводу дальнейших связей между классическими распределениями из класса Мейкснера и свободной теорией вероятностей см.…”