Generalized Meixner-type Free Gamma Distributions

• The paper presents that generalized Meixner-type free gamma distributions extend the free gamma law using an R-transform, establishing unique convolution and scaling properties.
• They are characterized analytically through explicit density formulas, free cumulants, and Cauchy transforms, which clarify conditions for selfdecomposability and unimodality.
• Their structural formulation unifies free gamma laws, scaled free beta prime laws, and free equilibrium measures, linking random matrix models with free entropy maximization.

Searching arXiv for the relevant papers and related terminology. The term generalized Meixner-type free gamma distributions refers to the three-parameter family

$$\mu_{t,\theta,\lambda}\qquad (t,\theta>0,\ \lambda\ge 1),$$

introduced as a class of probability measures on $\mathbb R_{\ge 0}$ that includes both the free gamma distributions introduced by Anshelevich and certain scaled free beta prime distributions introduced by Yoshida (Sakuma et al., 21 Aug 2025). In the source introducing this family, these laws are defined through an $R$-transform, placed inside the free Meixner framework, and related to Gibbs measures and free entropy maximization. At the same time, the literature contains another important notion of a free Gamma distribution, namely the Bercovici–Pata image $\nu_\alpha=\Lambda(\mu_\alpha)$ of the classical Gamma law, studied as a free selfdecomposable distribution rather than as a Meixner-class law (Haagerup et al., 2013). This distinction is central: generalized Meixner-type free gamma distributions belong to the Meixner-side theory, whereas the Bercovici–Pata free Gamma laws are analyzed through free infinite divisibility and analytic subordination, not through a generalized Meixner classification.

1. Definition and placement within the free Meixner framework

The family $\mu_{t,\theta,\lambda}$ is defined by its $R$-transform as

$$R_{\mu_{t,\theta,\lambda}}(z) =\int_{\mathbb R}\left(\frac{1}{1-zx}-1\right)\frac{t\,k_{\theta,\lambda}(x)}{x}\,dx, \qquad z\in \mathbb C^-,$$

where $k_{\theta,\lambda}$ is the Marchenko–Pastur density

$$k_{\theta,\lambda}(x) = \frac{\sqrt{(a^+-x)(x-a^-)}}{2\pi\theta x}\mathbf 1_{(a^-,a^+)}(x), \qquad a^\pm=\theta(\sqrt\lambda\pm 1)^2.$$

The same paper derives the closed form

$$R_{\mu_{t,\theta,\lambda}}(z) = t\cdot \frac{1+\theta(1-\lambda)z-\sqrt{(1+\theta(1-\lambda)z)^2-4\theta z}}{2\theta},$$

which makes the deformation parameter $\mathbb R_{\ge 0}$0 explicit (Sakuma et al., 21 Aug 2025).

The family is called generalized Meixner-type because it sits inside the free Meixner framework: $\mathbb R_{\ge 0}$1 where $\mathbb R_{\ge 0}$2 is the centered free Meixner distribution. In this formulation, $\mathbb R_{\ge 0}$3 is a shifted free Meixner law, with parameters belonging to a Meixner regime satisfying $\mathbb R_{\ge 0}$4 (Sakuma et al., 21 Aug 2025).

This placement is consistent with the broader characterization of free Meixner laws by Jacobi parameters that are constant from level $\mathbb R_{\ge 0}$5 onward,

$\mathbb R_{\ge 0}$6

together with the continued-fraction expansion of the Cauchy transform. Within that broader family, the literature explicitly lists free Gamma among the canonical subclasses, alongside the semicircle, Marchenko–Pastur, free Pascal, free binomial, and free hyperbolic secant laws (Lenczewski, 2013).

A further point of terminology comes from the characterization paper on free Meixner laws: there the normalized family $\mathbb R_{\ge 0}$7 is classified by the parameter relation $\mathbb R_{\ge 0}$8 as the free Gamma law, identifying free Gamma as a boundary case of the free Meixner class (Ejsmont, 2012). This suggests that, in the Meixner literature, “free Gamma” is a structural subclass of free Meixner laws, whereas in the Bercovici–Pata literature the same phrase designates a different construction.

2. Relation to earlier notions of free gamma distributions

For $\mathbb R_{\ge 0}$9, the generalized family reduces to the Meixner-type free gamma law: $R$0 The source describes $R$1 as Anshelevich’s free gamma distribution and records its $R$2-transform as

$R$3

It also satisfies the free analogues

$R$4

so the generalized family extends a free convolution semigroup already present in the Meixner-type gamma case (Sakuma et al., 21 Aug 2025).

By contrast, the paper “On the free Gamma distributions” defines, for each $R$5,

$R$6

where $R$7 is the Bercovici–Pata bijection and $R$8 is the classical Gamma law on $R$9 with density

$\nu_\alpha=\Lambda(\mu_\alpha)$0

That paper explicitly places $\nu_\alpha=\Lambda(\mu_\alpha)$1 in the framework of free selfdecomposable distributions and states that it does not identify $\nu_\alpha=\Lambda(\mu_\alpha)$2 as a Meixner law or as a member of a generalized Meixner family (Haagerup et al., 2013).

This terminological split matters. The generalized Meixner-type family $\nu_\alpha=\Lambda(\mu_\alpha)$3 extends the Anshelevich/Meixner-type free gamma law, not the Bercovici–Pata free Gamma law $\nu_\alpha=\Lambda(\mu_\alpha)$4. A plausible implication is that two different research programs use “free gamma” for structurally different objects: one governed by the Meixner $\nu_\alpha=\Lambda(\mu_\alpha)$5-transform algebra, the other by the Bercovici–Pata correspondence and free selfdecomposability.

3. Explicit transforms, densities, cumulants, and support

The Cauchy transform of $\nu_\alpha=\Lambda(\mu_\alpha)$6 is computed explicitly as

$\nu_\alpha=\Lambda(\mu_\alpha)$7

with

$\nu_\alpha=\Lambda(\mu_\alpha)$8

From this one obtains the absolutely continuous part

$\nu_\alpha=\Lambda(\mu_\alpha)$9

together with a possible atom at $\mu_{t,\theta,\lambda}$0: $\mu_{t,\theta,\lambda}$1 Thus the support changes qualitatively at the threshold $\mu_{t,\theta,\lambda}$2: below and at the threshold, the law is purely absolutely continuous on a compact interval; above it, an atom at the origin appears (Sakuma et al., 21 Aug 2025).

The paper also gives a free-cumulant formula. The first cumulant is

$\mu_{t,\theta,\lambda}$3

and for $\mu_{t,\theta,\lambda}$4,

$\mu_{t,\theta,\lambda}$5

The moments are then obtained by the moment-cumulant formula (Sakuma et al., 21 Aug 2025).

These formulas place the family in direct contact with standard free Meixner descriptions. For normalized free Meixner laws with $\mu_{t,\theta,\lambda}$6, $\mu_{t,\theta,\lambda}$7, the density is recorded in the free Meixner random-matrix paper as

$\mu_{t,\theta,\lambda}$8

on

$\mu_{t,\theta,\lambda}$9

with the possibility of one or two atoms outside the absolutely continuous part (Lenczewski, 2013). The generalized Meixner-type free gamma densities fit this larger pattern, but with the specific nonnegative-support structure encoded by $R$0.

4. Convolution, scaling, and mixture structure

The parameter $R$1 acts as a free-convolution time: $R$2 The paper also states the scaling relation

$R$3

so $R$4 behaves as a scale parameter (Sakuma et al., 21 Aug 2025).

For $R$5, the family admits a free multiplicative convolution formula: $R$6 where

$R$7

This exhibits $R$8 as a free multiplicative convolution of the free gamma law $R$9 with a Marchenko–Pastur law (Sakuma et al., 21 Aug 2025).

The same source identifies a beta-prime description: $$R_{\mu_{t,\theta,\lambda}}(z) =\int_{\mathbb R}\left(\frac{1}{1-zx}-1\right)\frac{t\,k_{\theta,\lambda}(x)}{x}\,dx, \qquad z\in \mathbb C^-,$$0 where

$$R_{\mu_{t,\theta,\lambda}}(z) =\int_{\mathbb R}\left(\frac{1}{1-zx}-1\right)\frac{t\,k_{\theta,\lambda}(x)}{x}\,dx, \qquad z\in \mathbb C^-,$$1

Equivalently, $$R_{\mu_{t,\theta,\lambda}}(z) =\int_{\mathbb R}\left(\frac{1}{1-zx}-1\right)\frac{t\,k_{\theta,\lambda}(x)}{x}\,dx, \qquad z\in \mathbb C^-,$$2 is a scaled free beta prime law (Sakuma et al., 21 Aug 2025).

At the boundary value

$$R_{\mu_{t,\theta,\lambda}}(z) =\int_{\mathbb R}\left(\frac{1}{1-zx}-1\right)\frac{t\,k_{\theta,\lambda}(x)}{x}\,dx, \qquad z\in \mathbb C^-,$$3

one has

$$R_{\mu_{t,\theta,\lambda}}(z) =\int_{\mathbb R}\left(\frac{1}{1-zx}-1\right)\frac{t\,k_{\theta,\lambda}(x)}{x}\,dx, \qquad z\in \mathbb C^-,$$4

and the paper states that at this boundary the measure becomes a free compound Poisson law (Sakuma et al., 21 Aug 2025). This boundary is therefore the transition point between purely absolutely continuous generalized gamma behavior and a regime in which a point mass at zero appears.

5. Selfdecomposability, unimodality, and entropy–potential correspondence

A sharp structural dichotomy is stated in the introduction paper for the generalized family: $$R_{\mu_{t,\theta,\lambda}}(z) =\int_{\mathbb R}\left(\frac{1}{1-zx}-1\right)\frac{t\,k_{\theta,\lambda}(x)}{x}\,dx, \qquad z\in \mathbb C^-,$$5 Thus only the undeformed free gamma case remains freely selfdecomposable. Likewise, for fixed $$R_{\mu_{t,\theta,\lambda}}(z) =\int_{\mathbb R}\left(\frac{1}{1-zx}-1\right)\frac{t\,k_{\theta,\lambda}(x)}{x}\,dx, \qquad z\in \mathbb C^-,$$6,

$$R_{\mu_{t,\theta,\lambda}}(z) =\int_{\mathbb R}\left(\frac{1}{1-zx}-1\right)\frac{t\,k_{\theta,\lambda}(x)}{x}\,dx, \qquad z\in \mathbb C^-,$$7

The unimodality threshold therefore coincides with the onset of the atom at zero (Sakuma et al., 21 Aug 2025).

The same work develops a potential correspondence based on Gibbs measures rather than the Bercovici–Pata bijection. The associated classical Gibbs law is

$$R_{\mu_{t,\theta,\lambda}}(z) =\int_{\mathbb R}\left(\frac{1}{1-zx}-1\right)\frac{t\,k_{\theta,\lambda}(x)}{x}\,dx, \qquad z\in \mathbb C^-,$$8

with

$$R_{\mu_{t,\theta,\lambda}}(z) =\int_{\mathbb R}\left(\frac{1}{1-zx}-1\right)\frac{t\,k_{\theta,\lambda}(x)}{x}\,dx, \qquad z\in \mathbb C^-,$$9

For $k_{\theta,\lambda}$0,

$k_{\theta,\lambda}$1

The paper emphasizes that this classical–free matching is not the Bercovici–Pata bijection, but a variational correspondence between equilibrium measures of entropy functionals under matching potentials (Sakuma et al., 21 Aug 2025).

Its main variational theorem states that for $k_{\theta,\lambda}$2 and

$k_{\theta,\lambda}$3

the measure $k_{\theta,\lambda}$4 is the unique maximizer of Voiculescu’s free entropy functional

$k_{\theta,\lambda}$5

over all probability measures on $k_{\theta,\lambda}$6 (Sakuma et al., 21 Aug 2025). This identifies the generalized Meixner-type free gamma distributions as free equilibrium measures for a natural logarithmic potential family.

For comparison, the Bercovici–Pata free Gamma laws $k_{\theta,\lambda}$7 are analyzed by entirely different methods. The central relation there is

$k_{\theta,\lambda}$8

with $k_{\theta,\lambda}$9 built from the Cauchy transform of the classical exponential law. The resulting density is analytic on $$k_{\theta,\lambda}(x) = \frac{\sqrt{(a^+-x)(x-a^-)}}{2\pi\theta x}\mathbf 1_{(a^-,a^+)}(x), \qquad a^\pm=\theta(\sqrt\lambda\pm 1)^2.$$0, supported on $$k_{\theta,\lambda}(x) = \frac{\sqrt{(a^+-x)(x-a^-)}}{2\pi\theta x}\mathbf 1_{(a^-,a^+)}(x), \qquad a^\pm=\theta(\sqrt\lambda\pm 1)^2.$$1, and unimodal, but the classification invoked is free selfdecomposability under $$k_{\theta,\lambda}(x) = \frac{\sqrt{(a^+-x)(x-a^-)}}{2\pi\theta x}\mathbf 1_{(a^-,a^+)}(x), \qquad a^\pm=\theta(\sqrt\lambda\pm 1)^2.$$2, not Meixner structure (Haagerup et al., 2013).

6. Characterizations, operator models, and conceptual scope

The free Meixner setting supplies both structural characterizations and random-matrix realizations. In the characterization paper, the normalized free Meixner laws $$k_{\theta,\lambda}(x) = \frac{\sqrt{(a^+-x)(x-a^-)}}{2\pi\theta x}\mathbf 1_{(a^-,a^+)}(x), \qquad a^\pm=\theta(\sqrt\lambda\pm 1)^2.$$3 have Cauchy transform

$$k_{\theta,\lambda}(x) = \frac{\sqrt{(a^+-x)(x-a^-)}}{2\pi\theta x}\mathbf 1_{(a^-,a^+)}(x), \qquad a^\pm=\theta(\sqrt\lambda\pm 1)^2.$$4

and $$k_{\theta,\lambda}(x) = \frac{\sqrt{(a^+-x)(x-a^-)}}{2\pi\theta x}\mathbf 1_{(a^-,a^+)}(x), \qquad a^\pm=\theta(\sqrt\lambda\pm 1)^2.$$5-transform

$$k_{\theta,\lambda}(x) = \frac{\sqrt{(a^+-x)(x-a^-)}}{2\pi\theta x}\mathbf 1_{(a^-,a^+)}(x), \qquad a^\pm=\theta(\sqrt\lambda\pm 1)^2.$$6

The classification recorded there includes the free Gamma law as the case

$$k_{\theta,\lambda}(x) = \frac{\sqrt{(a^+-x)(x-a^-)}}{2\pi\theta x}\mathbf 1_{(a^-,a^+)}(x), \qquad a^\pm=\theta(\sqrt\lambda\pm 1)^2.$$7

The central theorem states that free Meixner laws are characterized by a linear regression condition together with a conditional moment identity involving a third-degree polynomial. Since free Gamma is the boundary case $$k_{\theta,\lambda}(x) = \frac{\sqrt{(a^+-x)(x-a^-)}}{2\pi\theta x}\mathbf 1_{(a^-,a^+)}(x), \qquad a^\pm=\theta(\sqrt\lambda\pm 1)^2.$$8, the theorem specializes automatically to the free Gamma case (Ejsmont, 2012).

Independently, the random-matrix model paper constructs a $$k_{\theta,\lambda}(x) = \frac{\sqrt{(a^+-x)(x-a^-)}}{2\pi\theta x}\mathbf 1_{(a^-,a^+)}(x), \qquad a^\pm=\theta(\sqrt\lambda\pm 1)^2.$$9 block Gaussian Hermitian ensemble

$$R_{\mu_{t,\theta,\lambda}}(z) = t\cdot \frac{1+\theta(1-\lambda)z-\sqrt{(1+\theta(1-\lambda)z)^2-4\theta z}}{2\theta},$$0

with a degenerating block regime

$$R_{\mu_{t,\theta,\lambda}}(z) = t\cdot \frac{1+\theta(1-\lambda)z-\sqrt{(1+\theta(1-\lambda)z)^2-4\theta z}}{2\theta},$$1

and deterministic diagonal shift

$$R_{\mu_{t,\theta,\lambda}}(z) = t\cdot \frac{1+\theta(1-\lambda)z-\sqrt{(1+\theta(1-\lambda)z)^2-4\theta z}}{2\theta},$$2

Under the first partial trace, the moments converge to those of the free Meixner law associated with $$R_{\mu_{t,\theta,\lambda}}(z) = t\cdot \frac{1+\theta(1-\lambda)z-\sqrt{(1+\theta(1-\lambda)z)^2-4\theta z}}{2\theta},$$3; for ensembles of such matrices, the paper proves asymptotic conditional freeness with respect to the pair of partial traces $$R_{\mu_{t,\theta,\lambda}}(z) = t\cdot \frac{1+\theta(1-\lambda)z-\sqrt{(1+\theta(1-\lambda)z)^2-4\theta z}}{2\theta},$$4 (Lenczewski, 2013). Since free Gamma is explicitly listed as one of the Meixner subclasses, this model provides a random-matrix realization of the encompassing class in which generalized Meixner-type free gamma distributions reside.

Taken together, these works delineate a precise conceptual scope. The generalized distributions $$R_{\mu_{t,\theta,\lambda}}(z) = t\cdot \frac{1+\theta(1-\lambda)z-\sqrt{(1+\theta(1-\lambda)z)^2-4\theta z}}{2\theta},$$5 unify four structures already present in the literature: free gamma distributions in the Meixner sense, scaled free beta prime distributions, shifted centered free Meixner laws, and free equilibrium measures for explicit logarithmic potentials (Sakuma et al., 21 Aug 2025). At the same time, the phrase “free Gamma distribution” remains non-uniform across free probability. One line of work uses it for Meixner-type laws characterized by $$R_{\mu_{t,\theta,\lambda}}(z) = t\cdot \frac{1+\theta(1-\lambda)z-\sqrt{(1+\theta(1-\lambda)z)^2-4\theta z}}{2\theta},$$6-transforms, Jacobi parameters, and cubic conditional moments; another uses it for Bercovici–Pata images of classical Gamma laws with analytic densities on half-lines and asymptotic selfdecomposability properties (Haagerup et al., 2013). A plausible implication is that any discussion of “generalized Meixner-type free gamma distributions” must specify that it concerns the Meixner-type branch of the theory rather than the Bercovici–Pata free Gamma laws.