Interpolation of Chebyshev Polynomials and Interacting Fock Spaces

Abstract

We discover a family of probability measures μa, 0 < a ≤ 1, [Formula: see text] which contains the arcsine distribution (a = 1) and semi-circle distribution (a = 1/2). We show that the multiplicative renormalization method can be used to produce orthogonal polynomials, called Chebyshev polynomials with one parameter a, which reduce to Chebyshev polynomials of the first and second kinds when a = 1 and 1/2 respectively. Moreover, we derive the associated Jacobi–Szegö parameters. This one-parameter family of probability measures coincides with the vacuum distribution of the field operator of the interacting Fock spaces related to the Anderson model.

On the non-classical infinite divisibility of power semicircle distributions

“…The case θ = 0 is the semicircle distribution on (−σ, σ) and the right-boundary case θ = ∞ given by Poincaré's theorem is the classical Gaussian distribution. Other important families of compactly supported distributions which are useful in non-commutative probability and that include the arcsine and semicircle distributions are considered in Kubo, Kuo and Namli [29], [30] and references therein.…”

Generating Functions of Cauchy–stieltjes Type for Orthogonal Polynomials

“…where ρ is an analytic function around 0 with ρ(0) = 0 and ρ ′ (0) = 0. φ is of the form h(ρ(z)x) for some function h so that when Theorem 1.1 holds, we say that the multiplicative renormalization method applies with h. In [18], [19], the authors answered the following question: characterize the family of probability measures applicable with h(x) = (1 − x) −1 . The Jacobi-Szegö parameters (α n ) n , (ω n ) n are shown to be stationary sequences from rank n = 0, n = 1 respectively, a fact that characterizes the so-called free Meixner distributions.…”

“…They are for instance solutions of a quadratic regression problem in free probability [10] (see also [11] for another characterization). As a matter of fact, we shall use free probability theory to explain the occurrence of the free Meixner family in [18], [19]. The techniques we use not only match [18], [19] to [2], but also give an elegant and easy proof for the representations of the Voiculescu transforms for the free Meixner distributions.…”

“…As a matter of fact, we shall use free probability theory to explain the occurrence of the free Meixner family in [18], [19]. The techniques we use not only match [18], [19] to [2], but also give an elegant and easy proof for the representations of the Voiculescu transforms for the free Meixner distributions. These representations was already derived in [9] only for freely infinitely divisible laws from the free Meixner family while here, we deal with all the distributions.…”

Applicability of multiplicative renormalization method for a certain function

“…includes those distributions obtained in [2,5,7,9], in particular, the arcsine and semi-circle distributions. The purpose of the present paper is to solve the above problem for the case when h(x) is given by…”