2.1 Harmonic analysis in free probability

In this article, we employ the framework of free harmonic analysis as introduced by [Reference Bercovici and VoiculescuBV93] (see also Chapter 3 in [Reference Mingo and SpeicherMS17]). To compare with classical probability theory, infinite divisibility is often used since there is the Bercovici–Pata bijection, a mapping that relates classical infinitely divisible distributions to their free counterparts (see [Reference Barndorff-Nielsen and ThorbjørnsenBNT02, Reference Barndorff-Nielsen and ThorbjørnsenBNT06, Reference Bercovici and PataBP99] for details). From this perspective, we introduce the necessary tools for our analysis in the following sections.

A probability measure $\mu $ on $\mathbb {R}$ is called freely infinitely divisible if for any $n\in \mathbb {N}$ there exists a probability measure $\mu _{n}\in {\mathcal {P}}(\mathbb {R})$ such that

$$ \begin{align*} \mu = \underbrace{\mu_{n}\boxplus \dots \boxplus \mu_{n}}_{n \text{ times}}, \end{align*} $$

where $\boxplus $ denotes the free additive convolution, which can be defined as the distribution of sum of freely independent self-adjoint operators. In this case, $\mu _n\in {\mathcal {P}}(\mathbb {R})$ is uniquely determined for each $n\in \mathbb {N}$ . The freely infinite divisible distributions can be characterized as those admitting a Lévy–Khintchine representation in terms of R-transform, which is the free analog of the cumulant transform $C_{\mu }(z) := \log (\widehat {\mu }(z))$ , where $\widehat {\mu }$ is the characteristic function of $\mu \in {\mathcal {P}}(\mathbb {R})$ . This was originally established by Bercovici and Voiculescu in [Reference Bercovici and VoiculescuBV93] for all Borel probability measures. To explain it, we gather analytic tools for free additive convolution $\boxplus $ . In order to define the R-transform (or free cumulant transform) $R_\mu $ of $\mu \in {\mathcal {P}}(\mathbb {R})$ , we need to define its Cauchy–Stieltjes transform  $G_\mu $ :

$$ \begin{align*} G_\mu(z)=\int_{\mathbb{R}}\frac{1}{z-t}\,\mu(\mathrm{d} t), \qquad(z\in\mathbb{C}^+). \end{align*} $$

Note, in particular, that $\Im (G_\mu (z))<0$ for any z in $\mathbb {C}^+$ , and hence we may consider the reciprocal Cauchy transform $F_\mu \colon \mathbb {C}^{+}\to \mathbb {C}^{+}$ given by $F_{\mu }(z)=1/G_{\mu }(z)$ . For any $\mu \in {\mathcal {P}}(\mathbb {R})$ and any $\lambda>0,$ there exist positive numbers $\alpha ,\beta ,$ and M such that $F_{\mu }$ is univalent on the set $\Gamma _{\alpha ,\beta }:=\{z \in \mathbb {C}^{+} \,|\, \Im (z)>\beta , |\Re (z)|<\alpha \Im (z)\}$ and such that $F_{\mu }(\Gamma _{\alpha ,\beta })\supset \Gamma _{\lambda ,M}$ . Therefore, the right inverse $F^{\langle -1 \rangle }_{\mu }$ of $F_{\mu }$ exists on $\Gamma _{\lambda ,M}$ , and the R-transform (or free cumulant transform) $ R_\mu $ is defined by

$$ \begin{align*} R_{\mu}(w) =wF^{\langle -1 \rangle}_{\mu}\left(\frac{1}{w} \right)-1, \quad\text{for all } w \text{ such that } 1/w \in \Gamma_{\lambda,M}. \end{align*} $$

The free version of the Lévy–Khintchine representation now amounts to the statement that $\mu \in {\mathcal {P}}(\mathbb {R})$ is freely infinitely divisible if and only if there exist $a\ge 0$ , $\gamma \in \mathbb {R}$ and a Lévy measureFootnote ⁴ $\nu $ such that

(2.1) $$ \begin{align} R_{\mu}(w) = a w^{2}+\gamma w + \int_{\mathbb{R}}\left(\frac{1}{1- w x}-1-w x \mathbf{1}_{[-1,1]}(x)\right)\nu(\mathrm{d} x),\qquad w\in\mathbb{C}^-. \end{align} $$

The triplet $(a,\nu ,\gamma )$ is uniquely determined and referred to as the free characteristic triplet for $\mu $ , and the measure $\nu $ is referred to as the free Lévy measure for $\mu $ . Recently, free infinite divisibility has been proved for normal distributions [Reference Belinschi, Bożejko, Lehner and SpeicherBBLS11], some of the Boolean-stable distributions [Reference Arizmendi and HasebeAH14], some of the beta distributions, and some of the gamma distributions, including the chi-square distribution and powers of random variables distributed as these distributions [Reference HasebeHas14, Reference HasebeHas16] and generalized power distributions with free Poisson term [Reference Morishita and UedaMU20].

As one of the most important subclass of freely infinitely divisible distributions, we introduce the concepts of freely self-decomposable distributions. A probability measure $\mu $ on $\mathbb {R}$ is said to be freely self-decomposable if for any $c\in (0,1),$ there exists ${\mu _c \in \mathcal {P}(\mathbb {R})}$ such that $\mu = \mu _c \boxplus D_c(\mu )$ . It is easy to see that every freely self-decomposable distribution is freely infinitely divisible. Moreover, it is known that $\mu \in \mathcal {P}(\mathbb {R})$ is freely self-decomposable if and only if its free Lévy measure $\nu $ is formed by

$$ \begin{align*}\nu(\mathrm{d}x) = \frac{k(x)}{|x|} \mathbf{1}_{\mathbb{R}\setminus \{0\}}(x) \mathrm{d}x, \end{align*} $$

where a function k is nondecreasing on $(-\infty ,0)$ and nonincreasing on $(0,\infty )$ (see [Reference Barndorff-Nielsen and ThorbjørnsenBNT02] for details). Examples and properties of freely self-decomposable distributions were investigated by [Reference Hasebe, Sakuma and ThorbjørnsenHST19, Reference Hasebe and ThorbjørnsenHT16, Reference Hasebe and UedaHU23, Reference Maejima and SakumaMS23].

Let us consider $\mu \in \mathcal {P}(\mathbb {R}_{\ge 0}) \setminus \{\delta _0\}$ . We define the S-transform of $\mu $ by

$$ \begin{align*}S_\mu(z) := \frac{1+z}{z} \Psi_\mu^{\langle-1\rangle}(z), \qquad z \in \Psi_\mu (i \mathbb{C}^+), \end{align*} $$

where $\Psi _\mu $ is the moment generating function that is

$$ \begin{align*}\Psi_\mu(z):= \int_{\mathbb{R}_{>0}} \frac{xz}{1-xz} \mu(\mathrm{d}x), \qquad z\in \mathbb{C}\setminus \mathbb{R}_{\ge0}, \end{align*} $$

and $\Psi _\mu (i\mathbb {C}^+)$ is a region contained in the circle with diameter $(\mu (\{0\})-1,0)$ . One can see that

(2.2) $$ \begin{align} S_{D_c(\mu)}(z) = \frac{1}{c} S_\mu(z), \qquad c>0. \end{align} $$

For $\mu ,\nu \in \mathcal {P}(\mathbb {R}_{\ge 0})\setminus \{\delta _0\}$ , we obtain $S_{\mu \boxtimes \nu }=S_\mu S_\nu $ on the common domain in which three S-transforms are defined, where $\mu \boxtimes \nu $ is called the free multiplicative convolution, which is the distribution of multiplication $\sqrt {X}Y\sqrt {X}$ of freely independent positive random variables $X\sim \mu $ and $Y\sim \nu $ . It is known that

(2.3) $$ \begin{align} R_\mu(zS_\mu(z)) =z, \end{align} $$

for small enough z in a neighborhood of $(\mu (\{0\})-1,0)$ (see [Reference Bercovici and VoiculescuBV92, Reference Bercovici and VoiculescuBV93] for details).

Example 2.1 (Marchenko–Pastur distribution)

We define the function $k_{\theta ,\lambda }$ as

$$ \begin{align*}k_{\lambda,\theta}(x):= \frac{\sqrt{(a^+-x)(x-a^-)}}{2\pi \theta x} \mathbf{1}_{(a^-,a^+)}(x), \end{align*} $$

where $\lambda ,\theta>0$ and $a^\pm :=\theta (\sqrt {\lambda }\pm 1)^2$ , respectively. The Marchenko–Pastur law $\pi _{\lambda ,\theta }$ with the shape parameter $\lambda $ and the mean parameter $\theta $ is defined as the probability measure given by

$$ \begin{align*}\pi_{\lambda,\theta}(\mathrm{d}x) := \max\{0, 1-\lambda\}\delta_0 + k_{\lambda,\theta}(x)\mathrm{d}x. \end{align*} $$

It is known that

$$ \begin{align*}R_{\pi_{\lambda,\theta}}(z) = \frac{\theta\lambda z}{1-\theta z}, \qquad z\in \mathbb{C}^- \end{align*} $$

and

$$ \begin{align*}S_{\pi_{\lambda,\theta}}(z) = \frac{1}{\theta(\lambda +z)}, \qquad z \text{ in a neighborhood of } (-1+\pi_{\theta,\lambda}(\{0\}),0). \end{align*} $$

From the form of R-transform, we notice that, for any $\theta ,\lambda>0$ ,

$$ \begin{align*}\pi_{\lambda,\theta} =D_{\theta} (\pi_{1,1}^{\boxplus \lambda}). \end{align*} $$

According to [Reference Haagerup and SchultzHS07, Proposition 3.13], we have

(2.4) $$ \begin{align} S_{\mu^{\langle -1 \rangle}}(z) = \frac{1}{S_\mu(-z-1)}, \qquad z\in (-1,0). \end{align} $$

Example 2.2 (Free positive stable law with index $1/2$ )

By (2.4), for $\lambda \ge 1$ ,

$$ \begin{align*} S_{(\pi_{\lambda,\theta})^{\langle -1\rangle}}(z)= \frac{1}{S_{\pi_{\lambda,\theta}}(-z-1)} = \theta(\lambda-1-z), \qquad z\in (-1,0). \end{align*} $$

In particular, the measure $(\pi _{1,1})^{\langle -1\rangle }$ has a probability density $\frac {\sqrt {4x-1}}{2\pi x^2} \mathbf {1}_{(1/4,\infty )}(x) \mathrm {d} x$ and is well known as the free positive stable law with index $1/2$ , introduced by [Reference Bercovici and PataBP99].

2.2 Centered free Meixner distributions

In this section, we introduce the three-parameter family $\{\nu _{s,a,b}: s\ge 0, a \in \mathbb {R}, b\ge -1\} \subset {\mathcal {P}}(\mathbb {R})$ in which their Cauchy transform is given by

(2.5) $$ \begin{align} G_{\nu_{s,a,b}}(z) &= \cfrac{1}{z-\cfrac{s}{z-a-\cfrac{s+b}{z-a-\cfrac{s+b}{\ddots}}}}\end{align} $$

(2.6) $$ \begin{align} &= \frac{(s+2b)z+sa-s\sqrt{(z-a)^2-4(s+b)}}{2(bz^2+saz+s^2)}. \end{align} $$

The measure $\nu _{s,a,b}$ is called the centered free Meixner distribution. According to [Reference AnshelevichAns03, Reference Bożejko and BrycBB06, Reference Saitoh and YoshidaSY01], $\nu _{s,a,b}$ is freely infinitely divisible whenever $b\ge 0$ , and an integral representation for the R-transform of $\nu _{s, a,b}$ is given by

$$ \begin{align*}R_{\nu_{s, a,b}}(z) = \int_{\mathbb{R}} \frac{z^2}{1-zx}s \ w_{a,b}(x)\mathrm{d}x, \end{align*} $$

where

$$ \begin{align*}w_{a,b}(x) = \frac{1}{2\pi b}\sqrt{4b-(x-a)^2}\mathbf{1}_{[a-2\sqrt{b}, a+2\sqrt{b}]}(x) \end{align*} $$

is the density of Wigner’s semicircle law with mean $a\in \mathbb {R}$ and variance $b\ge 0$ . Note that, the above R-transform differs from [Reference AnshelevichAns03, Reference Bożejko and BrycBB06, Reference Saitoh and YoshidaSY01] by a factor of z. In this case, one can see that $\nu _{s,a,b}= \nu _{1,a,b}^{\boxplus s}$ . By [Reference Bożejko and BrycBB06, Equation (4)], the R-transform of $\nu _{s,a,b}$ admits the explicit form

(2.7) $$ \begin{align} R_{\nu_{s,a,b}}(z)=\frac{2sz^2}{1-az+\sqrt{(1-az)^2-4bz^2}}, \qquad b\neq 0 \end{align} $$

and in the case $b=0$ , it reduces to

$$ \begin{align*}R_{\nu_{s,a,0}}(z) = \frac{sz^2}{1-az}. \end{align*} $$

According to [Reference Bożejko and BrycBB06, Theorem 3.2], the centered free Meixner law $\nu _{1,a,b}$ coincides with one of the following measures:

• • the Wigner’s semicircle law if $a=b=0$ ;

• • the Marchenko–Pastur distribution if $b=0$ and $a\neq 0$ ;

• • the free Pascal (negative binomial) distribution if $b>0$ and $a^2>4b$ ;

• • the free gamma distribution if $b>0$ and $a^2=4b$ ;

• • the pure free Meixner distribution if $b>0$ and $a^2<4b$ ;

• • the free binomial distribution if $-\min \{\alpha , 1-\alpha \} \le b <0$ , where $\alpha =\int _{\mathbb {R}} x^2\ \nu _{1,a,b}(\mathrm { d} x)$ .

By (1.1) and (2.7), the Meixner-type free gamma distribution $\eta _{t,\theta }$ can be expressed in terms of the centered free Meixner law (the free gamma distribution in the sense described above) as

$$ \begin{align*}\eta_{t,\theta} = \nu_{t\theta^2, 2\theta, \theta^2} \boxplus \delta_{t\theta}, \qquad t,\theta>0. \end{align*} $$

More generally, we show that the generalized Meixner-type free gamma distribution can be represented as a centered free Meixner law under a shift (see Proposition 3.2).

2.3 Entropy functionals with potentials

Assume that V is a $C^1$ -potential function V satisfying

$$ \begin{align*}V(x) \ge (1+\delta)\log (x^2+1), \qquad x\in \mathbb{R}, \qquad \text{for some } \delta>0, \end{align*} $$

and $\mathcal {Z} := \int e^{-V(x)}\mathrm {d}x<\infty $ .

By the Lagrangian multiplier method, it is known that the Gibbs distribution $\frac {1}{\mathcal {Z}} \exp \{-V(x)\}$ is a unique probability density which maximizes the Shannon entropy associated with the potential function V:

$$ \begin{align*}H_V(p):= -\int p(x)\log p(x)\mathrm{d}x - \int V(x) p(x)\mathrm{d}x, \end{align*} $$

among all probability density functions p on $\mathbb {R}$ .

According to [Reference JohanssonJoh98], it is known that for the above potential function V, the free entropy functional (see [Reference VoiculescuVoi93])

$$ \begin{align*}\Sigma_V(\mu) := \iint \log |x-y| \mu (\mathrm{d}x) \mu (\mathrm{d}y) - \int V(x) \mu(\mathrm{d}x), \end{align*} $$

among all probability measures $\mu $ on $\mathbb {R}$ , is known to be finite and has a unique maximizer $\mu _V$ (namely, the equilibrium measure of $\Sigma _V$ ). The support of $\mu _V$ is compact. Moreover, $\mu _V$ satisfies the following equation:

$$ \begin{align*}\mathcal{H} \mu_V(x) = \frac{1}{2} V'(x), \qquad x\in \text{supp}(\mu_V), \end{align*} $$

where $\mathcal {H}\mu $ is the Hilbert transform of a probability measure $\mu $ on $\mathbb {R}$ , that is,

$$ \begin{align*}\mathcal{H}\mu (x) :=\text{p.v.} \int \frac{1}{x-y} \mu(\mathrm{d} y) = \lim_{\varepsilon \to 0} \left(\int_{-\infty}^{x-\varepsilon} + \int_{x+\varepsilon}^\infty\right)\frac{1}{x-y} \mu(\mathrm{d} y), \quad x\in \mathbb{R}, \end{align*} $$

see also [Reference Saff and TotikST97, p. 27, Theorem 1.3] and [Reference BianeBia03, Equation (3.4)].

In connection with the above discussion, Hasebe and Szpojankowski [Reference Hasebe and SzpojankowskiHS19] pointed out a correspondence between the measure that maximizes the Shannon entropy and the equilibrium measure of the free entropy, from the perspective of maximizing entropy functionals with a potential. We call it the potential correspondence Footnote ⁵ in this article. In [Reference Hasebe and SzpojankowskiHS19], it was observed that the potential correspondence maps the classical generalized inverse Gaussian (GIG) distributions to the free GIG distributions introduced in [Reference FèralFer06]. Moreover, this correspondence maps the normal distributions $N(\mu ,\sigma ^2)$ to Wigner’s semicircle laws $w_{\mu ,\sigma ^2}(x)\mathrm {d} x$ for $\mu \in \mathbb {R}$ and $\sigma>0$ , and the gamma distribution $\gamma _{\lambda ,\theta }$ to the Marchenko–Pastur distributions $\pi _{\lambda ,\theta }$ for $\theta>0$ and $\lambda \ge 1$ .

3.1 Relation with centered free Meixner distributions

We establish an important connection between the centered free Meixner distributions $\nu _{s,a,b}$ (defined in Section 2.2) and the generalized Meixner-type free gamma distributions $\mu _{t,\theta ,\lambda }$ .

Proposition 3.2 For $t,\theta>0$ and $\lambda \ge 1$ , we have

$$ \begin{align*} \mu_{t,\theta,\lambda} = \nu_{t\theta\lambda, \theta(\lambda+1), \theta^2\lambda} \boxplus \delta_t. \end{align*} $$

Proof A direct computation shows that

$$ \begin{align*} R_{\nu_{t\theta\lambda, \theta(\lambda+1), \theta^2\lambda} \boxplus \delta_t}(z) &= R_{{\nu_{t\theta\lambda, \theta(\lambda+1), \theta^2\lambda}} } (z) + tz\\ &=\int_{\mathbb{R}} \frac{z^2}{1-zx} \cdot t\theta\lambda \cdot w_{\theta(\lambda+1),\theta^2\lambda}(x)\mathrm{d}x + tz\\ &=\int_{\mathbb{R}} \frac{xz^2}{1-zx} \cdot t k_{\lambda,\theta}(x)\mathrm{d}x+tz\\ &=tz \int_{\mathbb{R}} \frac{1}{1-zx} k_{\lambda,\theta}(x)\mathrm{d}x\\ &=\int_{\mathbb{R}} \left(\frac{1}{1-zx}-1\right) \frac{tk_{\lambda,\theta}(x)}{x} \mathrm{d}x = R_{\mu_{t,\theta,\lambda}}(z), \end{align*} $$

as desired.

Remark 3.3 According to Proposition 3.2, the generalized Meixner-type free gamma distribution $\mu _{t,\theta ,\lambda }$ coincides, up to a shift, with the centered free Meixner distribution $\nu _{s,a,b}$ when the parameters satisfy

$$ \begin{align*}s=t\theta\lambda, \quad a=\theta(\lambda+1), \quad \text{and} \quad b=\theta^2\lambda. \end{align*} $$

In this setting, since $a^2\ge 4b$ , the measure $\nu _{s,a,b}$ is either the free Pascal distribution (when $a^2>4b$ ) or the free gamma distribution (when $a^2=4b$ ) (see [Reference Bożejko and BrycBB06, p. 65]). Consequently, the measure $\mu _{t,\theta ,\lambda }$ can be interpreted as a suitably shifted version of either the free Pascal or the free gamma distribution.

It follows from (2.7) and Proposition 3.2 that

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

Due to the above representation of R-transform, we can understand the second parameter $\theta $ for the measure $\mu _{t,\theta ,\lambda }$ . Since

$$ \begin{align*} R_{\mu_{t,\theta,\lambda}}(z) = \frac{1}{\theta} R_{\mu_{t,1,\lambda}}(\theta z) =\frac{1}{\theta} R_{D_\theta(\mu_{t,1,\lambda})}(z) = R_{D_\theta(\mu_{t,1,\lambda})^{\boxplus \frac{1}{\theta}}}(z), \end{align*} $$

we get

$$ \begin{align*}\mu_{t,\theta,\lambda}= D_\theta(\mu_{t,1,\lambda})^{\boxplus \frac{1}{\theta}}. \end{align*} $$

We will explain the meaning of the third parameter $\lambda $ in Theorem 4.3.

3.2 Density, atom, and moments

In this section, we investigate the density, atom, and moments of $\mu _{t,\theta ,\lambda }$ . Thanks to (2.6) and Proposition 3.2, it is straightforward to see that

$$ \begin{align*}G_{\mu_{t,\theta,\lambda}}(z)=\frac{(t+2\theta)z-t(t-\theta(\lambda-1)) -t \sqrt{(z-\alpha^-)(z-\alpha^+)}}{2\theta z (z+t(\lambda-1))}, \qquad z\in \mathbb{C}^+, \end{align*} $$

where

(3.2) $$ \begin{align} \alpha^\pm:= \theta(\lambda+1)+t \pm 2\sqrt{\theta\lambda(\theta+t)}. \end{align} $$

The Stieltjes-inversion formula (see [Reference SchmüdgenSch12, Theorem F.6]) implies that

(3.3) $$ \begin{align} \frac{\mathrm{d}\mu_{t,\theta,\lambda}}{\mathrm{d}x}(x) = \frac{t\sqrt{(x-\alpha^-)(\alpha^+-x)}}{2\pi \theta x (x +t(\lambda-1))} \mathbf{1}_{[\alpha^-,\alpha^+]}(x). \end{align} $$

Since

$$ \begin{align*} \lim_{z\to 0} z G_{\mu_{t,\theta,\lambda}}(z) = \frac{-t(t-\theta(\lambda-1)) + t|t-\theta(\lambda-1)|}{2t\theta(\lambda-1)}, \end{align*} $$

we have

(3.4) $$ \begin{align} \mu_{t,\theta,\lambda}(\{0\}) = \begin{cases} 0, & 1 \le \lambda \le 1+t/\theta\\ \\ 1-\dfrac{t}{\theta(\lambda-1)}, & \lambda> 1+t/\theta. \end{cases} \end{align} $$

In particular, $\mu _{t,\theta ,\lambda }$ has no singular continuous part since it is freely infinitely divisible (see [Reference Belinschi and BercoviciBB04, Theorem 3.4]). We summarize the above result as follows.

Proposition 3.4 (Density and atom)

For $t,\theta>0$ and $\lambda \ge 1$ , we get

$$ \begin{align*}\mu_{t,\theta,\lambda}(\mathrm{d} x) = \max\left\{0, 1-\dfrac{t}{\theta(\lambda-1)}\right\} \delta_0 (\mathrm{d} x) + \frac{t\sqrt{(x-\alpha^-)(\alpha^+-x)}}{2\pi \theta x (x +t(\lambda-1))} \mathbf{1}_{[\alpha^-,\alpha^+]}(x)\mathrm{d} x. \end{align*} $$

Next, we compute the moments of $\mu _{t,\theta ,\lambda }$ :

$$ \begin{align*}m_n(\mu_{t,\theta,\lambda})= \int_{\mathbb{R}} x^n \mu_{t,\theta,\lambda}(\mathrm{d}x), \qquad n\ge 1. \end{align*} $$

To obtain $m_n(\mu _{t,\theta ,\lambda })$ , we first compute its n-th free cumulant $\kappa _n(\mu _{t,\theta ,\lambda })$ , which is defined as the coefficient of $z^n$ in the power series expansion of the R-transform $R_{\mu _{t,\theta ,\lambda }}(z)$ . By the computation in the proof of Proposition 3.2, we have

$$ \begin{align*} R_{\mu_{t,\theta,\lambda}}(z) &=tz \int_{\mathbb{R}} \frac{1}{1-zx} k_{\lambda,\theta}(x)\mathrm{d} x = \sum_{n=0}^\infty tm_n(\pi_{\lambda,\theta})z^{n+1}, \end{align*} $$

where $m_0(\pi _{\lambda ,\theta })=1$ . Comparing the coefficients of $z^n$ then yields the following result:

(3.5) $$ \begin{align} \kappa_1(\mu_{t,\theta,\lambda}) &= t; \end{align} $$

(3.6) $$ \begin{align} \kappa_{n+1}(\mu_{t,\theta,\lambda}) &= t m_n(\pi_{\lambda,\theta}) = \frac{t\theta^{n}}{n} \sum_{k=0}^{n-1} \binom{n}{k}\binom{n}{k+1} \lambda^k, \qquad n\ge1. \end{align} $$

Proposition 3.5 (Moments)

Consider $t,\theta>0$ and $\lambda \ge 1$ . Then, $m_1(\mu _{t,\theta ,\lambda })=t$ and for $n\ge 2$ ,

$$ \begin{align*} m_n(\mu_{t,\theta,\lambda})=\sum_{m=1}^n \sum_{\substack{r_1,\dots, r_n\ge 0\\ r_1+\cdots+ r_n=m \\ r_1+ 2 r_2 + \cdots + nr_n=n}}P_m^{(n)}(r_1,\dots, r_n)t^m \theta^{n-m} \prod_{s=1}^{n-1} \left( \frac{1}{s}\sum_{k=0}^{s-1} \binom{s}{k}\binom{s}{k+1}\lambda^{k}\right)^{r_{s+1}}, \end{align*} $$

where

$$ \begin{align*}P_m^{(n)}(r_1,\dots, r_n):=\frac{n!}{r_1!r_2!\cdots r_n! ( n - m + 1)!}. \end{align*} $$

Proof One can observe that $m_1(\mu _{t,\theta ,\lambda })=\kappa _1(\mu _{t,\theta ,\lambda })=t$ by (3.5). Let us consider ${n\ge 2}$ . It is known that the number of non-crossing partitions with $r_1$ blocks of size $1$ , $r_2$ blocks of size $2$ , $\dots $ , $r_n$ blocks of size n equals $P_m^{(n)}(r_1,\dots , r_n)$ , where $r_1+r_2+\cdots + r_n=m$ . By the moment-cumulant formula (cf. [Reference Nica and SpeicherNS06, Proposition 11.4]) and (3.6), for $n\ge 2$ , we obtain

$$ \begin{align*} &m_n(\mu_{t,\theta,\lambda}) = \sum_{\pi \in \mathcal{NC}(n)} \prod_{V\in \pi} \kappa_{|V|}(\mu_{t,\theta,\lambda})\\ &\quad=\sum_{m=1}^n\sum_{\substack{r_1,\dots, r_n\ge 0\\ r_1+\cdots+ r_n=m \\ r_1+ 2 r_2 + \cdots + nr_n=n}}P_m^{(n)}(r_1,\dots, r_n)\kappa_1(\mu_{t,\theta,\lambda})^{r_1}\kappa_2(\mu_{t,\theta,\lambda})^{r_2}\dots \kappa_n(\mu_{t,\theta,\lambda})^{r_n}\\ &\quad=\sum_{m=1}^n \sum_{\substack{r_1,\dots, r_n\ge 0\\ r_1+\cdots+ r_n=m \\ r_1+ 2 r_2 + \cdots + nr_n=n}}P_m^{(n)}(r_1,\dots, r_n) t^m \theta^{n-m} \prod_{s=1}^{n-1} \left( \frac{1}{s}\sum_{k=0}^{s-1} \binom{s}{k}\binom{s}{k+1}\lambda^{k}\right)^{r_{s+1}}. \end{align*} $$

In Section 4.2, we will notice that the measure $\mu _{t,\theta ,\lambda }$ coincides with a certain scaled free beta prime distribution introduced by [Reference YoshidaYos20] for $t,\theta>0$ and $\lambda>1$ . According to [Reference YoshidaYos20, Theorem 6.1], another combinatorial representation of $m_n(\mu _{t,\theta ,\lambda })$ will be obtained (see Corollary 4.4 later).

3.3 Free self-decomposability and unimodality

One can easily see free self-decomposability for the measure $\mu _{t,\theta ,\lambda }$ .

A probability measure $\mu $ on $\mathbb {R}$ is said to be unimodal if there exist $a\in \mathbb {R}$ and a density function f, which is nondecreasing on $(-\infty , a)$ and nonincreasing on $(a,\infty )$ , such that

$$ \begin{align*}\mu (\mathrm{d} x) = \mu (\{a\}) \delta_a + f(x) \mathrm{d}x. \end{align*} $$

According to [Reference Hasebe and ThorbjørnsenHT16, Theorem 1], every freely self-decomposable distribution is unimodal. Hence, $\mu _{t,\theta ,1}$ is unimodal for any $t,\theta>0$ by Proposition 3.7. For given $t,\theta>0$ , we investigate the values of $\lambda $ for which $\mu _{t,\theta ,\lambda }$ remains unimodal.

Proof If $1\le \lambda < 1+t/\theta $ , we get $\mu _{t,\theta ,\lambda }(\mathrm {d}x) = \frac {t}{2\pi \theta } f(x)dx$ by (3.3), where

$$ \begin{align*}f(x)=\frac{\sqrt{(x-\alpha^-)(\alpha^+-x)}}{x(x+t(\lambda-1))}, \qquad x\in (\alpha^-,\alpha^+), \end{align*} $$

and $\alpha ^\pm $ is defined by (3.2). By elementary calculus, we obtain

$$ \begin{align*}f'(x)=\frac{k(x)}{2x^2(x+t(\lambda-1))^2\sqrt{(x-\alpha^-)(\alpha^+-x)}}, \end{align*} $$

where

$$ \begin{align*}k(x):=2x^3 -3(\alpha^++\alpha^-)x^2+\{4\alpha^+\alpha^- -t(\lambda-1)(\alpha^++\alpha^-)\}x+2\alpha^+\alpha^-t(\lambda-1). \end{align*} $$

We show that there exists a unique solution $x \in (\alpha ^-,\alpha ^+)$ of the equation $k(x)=0$ . Since

$$ \begin{align*} k(\alpha^-)&=\{\alpha^- +t (\lambda-1) \}\alpha^-(\alpha^+-\alpha^-)>0,\\ k(\alpha^+)&=\{\alpha^+ +t (\lambda-1) \}\alpha^+(\alpha^--\alpha^+)<0, \end{align*} $$

it follows from the intermediate value theorem that there exists at least one solution $x\in (\alpha ^-,\alpha ^+)$ to the equation $k(x)=0$ . Next, we establish the uniqueness of solutions to $k(x)=0$ . To this end, we show that the function $k(x)$ is monotone on the interval $(\alpha ^-, \alpha ^+)$ . Since

$$ \begin{align*} k'(x) &= 6x^2-6(\alpha^++\alpha^-) x + \{4\alpha^+\alpha^- - t(\lambda-1)(\alpha^++\alpha^-)\}\\ &= 6\left(x-\frac{\alpha^++\alpha^-}{2}\right)^2 -\frac{3}{2}(\alpha^++\alpha^-)^2 + 4\alpha^+\alpha^- -t(\lambda-1)(\alpha^++\alpha^-)\\ &\le 6\left(\alpha^+-\frac{\alpha^++\alpha^-}{2}\right)^2 -\frac{3}{2}(\alpha^++\alpha^-)^2 + 4\alpha^+\alpha^- -t(\lambda-1)(\alpha^++\alpha^-)\\ &=-2\alpha^+\alpha^- - t(\lambda-1)(\alpha^++\alpha^-)<0, \end{align*} $$

the function $k(x)$ is strictly decreasing on $(\alpha ^-,\alpha ^+)$ . Hence, the equation $k(x)=0$ has a unique solution in $(\alpha ^-,\alpha ^+)$ , denoted by $x_0\in (\alpha ^-,\alpha ^+)$ . Consequently, the function $f(x)$ is strictly increasing on $(\alpha ^-,x_0)$ and strictly decreasing on $(x_0,\alpha ^+)$ , implying that $\mu _{t,\theta ,\lambda }$ is unimodal with mode $x_0$ .

If $\lambda = 1+t/\theta $ , then $\alpha ^-=0$ and $\alpha ^+=4(\theta +t)$ . In this case, one can verify that $f'(x)<0$ for all $x\in (0,\alpha ^+)$ . Hence, $\mu _{t,\theta ,1+t/\theta }$ is unimodal with mode $0$ .

If $\lambda> 1+t/\theta $ , then $\mu _{t,\theta ,\lambda }$ has an atom at $0$ by (3.4). However, the absolutely continuous part of $\mu _{t,\theta ,\lambda }$ possesses a mode $x_0$ , as shown above. Therefore, in this case, $\mu _{t,\theta ,\lambda }$ is not unimodal.

3.4 Background driving free Lévy process

Let $\mu $ be a freely self-decomposable distribution on $\mathbb {R}$ . By [Reference Barndorff-Nielsen and ThorbjørnsenBNT06, Theorem 6.5], there exists a free Lévy processFootnote ⁶ $\{Z_t\}_{t\ge 0}$ affiliated with some $W^\ast $ -probability space such that

$$ \begin{align*}\mu = \mathcal{L}\left( \int_0^\infty e^{-t} \mathrm{d}Z_t \right) \end{align*} $$

and the free Lévy measure $\nu $ of the law $\mathcal {L}(Z_1)$ satisfies

$$ \begin{align*}\int_{\mathbb{R}\setminus[-1,1]} \log(1+|x|) \nu(\mathrm{d}x) <\infty, \end{align*} $$

where $\int _0^\infty e^{-t} \mathrm {d}Z_t$ is the free stochastic integral with respect to $\{Z_t\}_{t\ge 0}$ (see [Reference Barndorff-Nielsen and ThorbjørnsenBNT06, Section 6] and [Reference Maejima and SakumaMS23] for details). The free Lévy process $\{Z_t\}_{t\ge 0}$ is called the background driving free Lévy process of $\mu $ .

By Proposition 3.7, the measure $\mu _{1,\theta ,1}$ is freely self-decomposable. From the construction above, we can then consider the background driving free Lévy process $\{Z_t\}_{t\ge 0}$ of $\mu _{1,\theta ,1}$ .

Proof (1) Recall that

$$ \begin{align*}R_{\mu_{1,\theta,1}}(z) = \frac{1-\sqrt{1-4\theta z}}{2\theta}, \qquad z\in \mathbb{C}^-. \end{align*} $$

By [Reference Maejima and SakumaMS23, Theorem 6.7], we have

$$ \begin{align*} R_{Z_t}(z) = tz \frac{d}{dz} R_{\mu_{1,\theta,1}}(z) = \frac{tz}{\sqrt{1-4\theta z}}. \end{align*} $$

(2) We can further compute the R-transform of $Z_t$ as follows:

$$ \begin{align*} R_{Z_t}(z) &= \frac{tz}{\sqrt{1-4\theta z}}=-\frac{t}{2\sqrt{\theta}} \frac{-z}{\sqrt{-z+\frac{1}{4\theta}}}\\ &=-\frac{t}{2\sqrt{\theta}} \int_{\frac{1}{4\theta}}^\infty \frac{-z}{-z+x} \cdot \frac{1}{\pi \sqrt{x-\frac{1}{4\theta}}}\mathrm{d}x = \int_0^{4\theta} \kern-1.5pt\left( \frac{1}{1-zx}-1\right) \frac{t}{\pi x \sqrt{x(4\theta -x)}}\mathrm{d}x, \end{align*} $$

where the third equality follows from [Reference Schilling, Song and VondracekSSV12, p. 304] or [Reference Maejima and SakumaMS23, Example 7.2]. Consequently, the Lévy measure of $Z_t$ is $\frac {t}{\pi x \sqrt {x(4\theta -x)}} \mathbf {1}_{(0,4\theta )}(x)\mathrm {d}x$ .

The regularity properties of the law of $Z_t$ can be analyzed as follows.

Proof Let $\nu _t$ be the free Lévy measure of $\mathcal {L}(Z_t)$ . Due to Lemma 3.10(2), we have

$$ \begin{align*} \nu_t(\mathbb{R}) &=\frac{t}{\pi} \int_0^{4\theta} x^{-\frac{3}{2}} (4\theta- x)^{-\frac{1}{2}} \mathrm{d}x = \infty. \end{align*} $$

According to [Reference Hasebe and SakumaHS17, Theorem 3.4], the measure $\mathcal {L}(Z_t)$ is absolutely continuous with respect to Lebesgue measure with continuous density on $\mathbb {R}$ .

Further, we can obtain the n-th free cumulant of the law of $Z_t$ .

Proposition 3.13 For $n\ge 1$ , we have

$$ \begin{align*}\kappa_1(Z_t)= t \qquad \text{and} \qquad \kappa_n(Z_t) = t(2\theta)^{n-1} \frac{(2n-3)!!}{(n-1)!} \quad \text{for} \quad n\ge 2. \end{align*} $$

Proof By Lemma 3.10(2), for z small enough (more strictly, $|z|<1/4\theta $ ), we have

$$ \begin{align*} R_{Z_t}(z) &= z \int_0^{4\theta} \frac{1}{1-zx} \frac{t}{\pi\sqrt{x(4\theta-x)}}\mathrm{d}x\\ &=z \int_0^1 \frac{1}{1-4\theta z u} \cdot \frac{t}{\pi\sqrt{u(1-u)}}\mathrm{d}u \qquad {(x=4\theta u)}\\ &=\frac{tz}{\pi} \int_0^1 u^{-\frac{1}{2}} (1-u)^{-\frac{1}{2}}(1-4\theta zu)^{-1}\mathrm{d}u\\ &=\frac{tz}{\pi} \frac{\Gamma(\frac{1}{2})^2}{\Gamma(1)} {}_2 F_1\left(1,\frac{1}{2}; 1; 4\theta z \right) \qquad \text{(by Euler integral representation)}\\ &=tz \sum_{n=0}^\infty \frac{(1)^{(n)}(\frac{1}{2})^{(n)}}{(1)^{(n)} n!} (4\theta z)^n \qquad {((x)^{(n)}:=x(x+1)\dots (x+n-1))}\\ &=tz + \sum_{n=2}^\infty t (2\theta)^{n-1}\frac{(2n-3)!!}{(n-1)!} z^n. \end{align*} $$

Comparing the coefficients of $z^n$ then yields the desired result.

3.5 Correlation of a free gamma process

In noncommutative setting, we can consider covariance and correlation as follows. Let $(\mathcal {A},\varphi )$ be a $C^\ast $ -probability space and $x,y\in \mathcal {A}$ . Then, their covariance is defined by

$$ \begin{align*}\text{Cov}(x,y) := \varphi(xy)-\varphi(x)\varphi(y). \end{align*} $$

It is easy to see that $\text {Cov}(x,y)=0$ if $x,y$ are free. Next, their correlation can be defined by

$$ \begin{align*}\text{Corr}(x,y) := \frac{\text{Cov}(x,y)}{\sqrt{\kappa_2(x)}\sqrt{\kappa_2(y)}}, \end{align*} $$

when $x,y$ have nonzero second free cumulant (variance). In general, we note that $\text {Corr}(x,y) \neq \text {Corr}(y,x)$ since $xy\neq yx$ .

Let $\{X_t\}_{t\ge 0}$ be a stochastic process in a $C^\ast $ -probability space. The process $\{X_t\}_{t\ge 0}$ is called a Meixner-type free gamma process if it is a free Lévy process whose marginal distribution at time $1$ is $\mu _{1,\theta ,\lambda }$ for some $\theta>0$ and $\lambda \ge 1$ . By the definition of free Lévy processes, it follows that $X_t\sim \mu _{t,\theta ,\lambda }$ for $t>0$ . Below, we compute the correlation of a free gamma process.

Proposition 3.14 (Correlation)

Let $\{X_t\}_{t\ge 0}$ be a Meixner-type free gamma process in a $C^\ast $ -probability space $(\mathcal {A},\varphi )$ . For any $s,t>0$ , we have

$$ \begin{align*} \text{Corr}(X_s, X_t)= \text{Corr}(X_t,X_s)= \sqrt{\frac{s}{t}}. \end{align*} $$

Proof For $s<t$ , we have

$$ \begin{align*} \text{Cov}(X_s,X_t) &= \varphi(X_s X_t) - \varphi(X_s)\varphi(X_t)\\ &= \varphi(X_s (X_t-X_s) + X_s^2) -\varphi(X_s)\varphi(X_t)\\ &= \varphi(X_s (X_t-X_s)) + \varphi(X_s^2)- \varphi(X_s)\varphi(X_t). \end{align*} $$

By Theorem 3.5 (or Example 3.6), we have $\varphi (X_s)=s$ and $\varphi (X_s^2)=s^2+\theta s$ . Since $\{X_t\}_{t\ge 0}$ is a free Lévy process, $X_s$ and $X_t-X_s$ are free, and hence $\varphi (X_s(X_t-X_s))=\varphi (X_s) \varphi (X_t-X_s)$ . Moreover, by the definition of a free Lévy process, $X_t-X_s \overset {\mathrm {d}}{=} X_{t-s}$ . Finally, we get

$$ \begin{align*} \text{Cov}(X_s,X_t) = s(t-s) + s^2+\theta s -st =\theta s. \end{align*} $$

By (3.5) (or Example 3.6 again), we obtain $\kappa _2(X_s)=\kappa _2(\mu _{s,\theta ,\lambda })=\theta s$ . Hence,

$$ \begin{align*}\text{Corr}(X_s,X_t) = \frac{\theta s} {\sqrt{\theta s} \sqrt{\theta t}} = \sqrt{\frac{s}{t}}. \end{align*} $$

Recalling the computation of their covariance, we observe that $\text {Cov}(X_t,X_s)=\text {Cov}(X_s,X_t)$ even if $s<t$ . Consequently, it also follows that $\text {Corr}(X_t,X_s)=\text {Corr}(X_s,X_t)$ .

If $s=t$ , then

$$ \begin{align*}\text{Cov}(X_s,X_s)=\varphi(X_s^2)-\varphi(X_s)^2=s^2+\theta s - s^2 = \theta s, \end{align*} $$

and therefore $\text {Corr}(X_s,X_s)=1$ .

In classical probability, it is known that the correlation of the gamma process $\{G_t\}_{t\ge 0}$ is

$$ \begin{align*}\text{Corr}(G_s,G_t) = \sqrt{\frac{s}{t}} \qquad \text{for} \qquad s,t>0. \end{align*} $$

For the reason, Proposition 3.14 is entirely analogous to the above classical result.

4.1 S-transform

In this section, we compute the S-transform of $\mu _{t,\theta ,\lambda }$ .

Lemma 4.1 For $t,\theta>0$ and $\lambda \ge 1$ , we have

$$ \begin{align*}S_{\mu_{t,\theta,\lambda}} (z) = \frac{t-\theta z}{t(t+\theta(\lambda-1)z)}, \qquad z \in \left( -1 + \mu_{t,\theta,\lambda}(\{0\}),0 \right). \end{align*} $$

Proof A straightforward computation together with (2.3) and (3.1) shows that the compositional inverse of $R_{\mu _{t,\theta ,\lambda }} $ is

$$ \begin{align*}R_{\mu_{t,\theta,\lambda}}^{\langle -1 \rangle} (z) = \frac{z(\theta z-t)}{\theta t (1-\lambda)z -t^2}, \qquad z\in (-1+ \mu_{t,\theta,\lambda}(\{0\}),0), \end{align*} $$

which in turn implies that

$$ \begin{align*} S_{\mu_{t,\theta,\lambda}} (z) = \frac{R_{\mu_{t,\theta,\lambda}}^{\langle -1 \rangle} (z)}{z}= \frac{t-\theta z}{t(t+\theta (\lambda-1)z)}.\\[-42pt] \end{align*} $$

In particular, we show that the measure $\mu _{t,\theta ,1}$ is the inverse of Marchenko–Pastur distribution.

Proof The desired formula follows from the S-transform of $(\pi _{1+t/\theta , \theta /t^2})^{\langle -1\rangle }$ . Actually, from Example 2.2 and Lemma 4.1, we have

$$ \begin{align*} S_{(\pi_{1+t/\theta, \theta/t^2})^{\langle -1\rangle}}(z) = \frac{\theta}{t^2}\left(\frac{t}{\theta}-z\right) = S_{\mu_{t,\theta,1}}(z).\\[-42pt] \end{align*} $$

4.2 Free convolution formula

Using the S-transform, we can analyze the effect of the third parameter $\lambda $ on the measure $\mu _{t,\theta ,\lambda }$ as follows.

Theorem 4.3 (Free convolution formula for $\mu _{t,\theta ,\lambda }$ )

Consider $t,\theta>0$ . If $\lambda>1$ , then

(4.1) $$ \begin{align} \mu_{t,\theta,\lambda} &=D_{t(\lambda-1)}\left(\pi_{q,1} \boxtimes (\pi_{1+\frac{t}{\theta},1})^{\langle -1\rangle} \right) \end{align} $$

(4.2) $$ \begin{align} &= \mu_{t,\theta,1} \boxtimes \pi_{q,q^{-1}}, \end{align} $$

where $q = \frac {t}{\theta (\lambda -1)}$ . In particular, $\mu _{t,\theta ,1+t/\theta }= \mu _{t,\theta ,1} \boxtimes \pi _{1,1}$ . Hence, the measure $\mu _{t,\theta ,1+t/\theta }$ belongs to the class of free compound Poisson distributions.

Proof By Examples 2.1 and 2.2 and Lemma 4.1, we have

$$ \begin{align*} S_{ D_{t(\lambda-1)}\left(\pi_{q,1} \boxtimes (\pi_{1+\frac{t}{\theta},1})^{\langle -1\rangle} \right)}(z) &=\frac{1}{t(\lambda-1)} \frac{1}{\frac{t}{\theta(\lambda-1)}+z} \left(\frac{t}{\theta}-z\right)\\ &=\frac{t-\theta z}{t(t+\theta(\lambda-1)z)}\\ &=S_{\mu_{t,\theta,\lambda}}(z). \end{align*} $$

Thus, equation (4.1) holds. Since $\pi _{\lambda ,\theta }=D_\theta (\pi _{\lambda ,1})$ , we obtain

$$ \begin{align*} \mu_{t,\theta,\lambda} &=D_{t(\lambda-1)}\left(\pi_{\frac{t}{\theta(\lambda-1)},1} \boxtimes (\pi_{1+\frac{t}{\theta},1})^{\langle -1\rangle} \right) \\ &=D_{t(\lambda-1)} \circ D_{\frac{t}{\theta(\lambda-1)}} \left( \pi_{\frac{t}{\theta(\lambda-1)},\frac{\theta(\lambda-1)}{t}} \boxtimes (\pi_{1+\frac{t}{\theta},1})^{\langle -1\rangle} \right)\\ &=\pi_{q,q^{-1}}\boxtimes (\pi_{1+\frac{t}{\theta},\frac{\theta}{t^2}})^{\langle-1\rangle}\\ &=\pi_{q,q^{-1}}\boxtimes \mu_{t,\theta,1}, \end{align*} $$

where the last equality follows from Lemma 4.2.

According to Theorem 4.3, for any $t,\theta>0$ and $\lambda>1$ , the measure $\mu _{t,\theta ,\lambda }$ coincides with

(4.3) $$ \begin{align} D_{t(\lambda-1)}\left(f\beta' \left(\frac{t}{\theta(\lambda-1)}, 1+\frac{t}{\theta} \right)\right), \end{align} $$

where

$$ \begin{align*}f\beta'(a, b): = \pi_{a,1}\boxtimes \pi_{b,1}^{\langle -1 \rangle}, \qquad a>0 \text{ and } b>1, \end{align*} $$

is the free beta prime distribution, introduced by Yoshida [Reference YoshidaYos20, Section 3.4]. Thus, by using [Reference YoshidaYos20, Theorem 6.1], we obtain a combinatorial formula for the moments of $\mu _{t,\theta ,\lambda }$ .

Corollary 4.4 For $t,\theta>0$ and $\lambda>1$ , the n-th moment of $\mu _{t,\theta ,\lambda }$ is given by

$$ \begin{align*}m_n(\mu_{t,\theta,\lambda}) = \theta^n \sum_{\pi \in \mathcal{NCL}(n)}\lambda^{|\pi|-\text{sg}(\pi)} \left(\frac{t}{\theta}\right)^{|\pi|-\text{dc}(\pi)}, \end{align*} $$

where $\mathcal {NCL}(n)$ is the set of all non-crossing linked partitions of $\{1,\dots ,n\}$ (see [Reference DykemaDyk07]), $\mathrm {sg}(\pi )$ is the number of singletons in $\pi $ , and $\mathrm {dc}(\pi )$ is the number of doubly covered elements by $\pi $ (see [Reference YoshidaYos20, Definition 5.7]).

4.3 Free beta prime distributions

In the previous section, we observed that $\mu _{t,\theta ,\lambda }$ is a suitably scaled free beta prime distribution when $\lambda>1$ . Using the fact, we now investigate the free beta prime distribution $f\beta '(a,b)$ for any $a>0$ and $b>1$ . A straightforward computation of the S-transform yields the following formula.

Proposition 4.5 For any $a>0$ and $b>1$ , we have

$$ \begin{align*}f\beta'(a,b)= \mu_{\frac{a}{b-1}, \frac{a}{(b-1)^2}, \frac{a+b-1}{a}}. \end{align*} $$

Proof It is easy to see that

(4.4) $$ \begin{align} S_{f\beta'(a,b)}(z)= \frac{b-1-z}{a+z} \qquad a>0, \ b>1. \end{align} $$

To determine $t,\theta>0$ and $\lambda>1$ from $a>0$ and $b>1$ , we compare the following equation:

$$ \begin{align*}\frac{b-1-z}{a+z} = \frac{t-\theta z}{t(t+\theta(\lambda-1)z)}, \end{align*} $$

where the RHS is the S-transform of some $\mu _{t,\theta ,\lambda }$ . Equivalently,

$$ \begin{align*} t(\lambda-1)=1, \quad -\theta a + t = -t^2 +t \theta(\lambda-1)(b-1) \quad \text{and} \quad ta = t^2(b-1). \end{align*} $$

Thus, one can see that $t=\frac {a}{b-1}$ , $\theta = \frac {a}{(b-1)^2}$ , and $\lambda =\frac {a+b-1}{a}$ , as desired.

From Proposition 4.5, we can identify analytic properties of the free beta prime distribution that were not discussed in [Reference YoshidaYos20].

Proof The free Lévy measure of $f\beta '(a,b)$ follows directly from the definition of $\mu _{t,\theta ,\lambda }$ and Proposition 4.5. Since $\frac {a+b-1}{a}>1$ , Proposition 3.7 together with Proposition 4.5 implies that $f\beta '(a,b)= \mu _{\frac {a}{b-1}, \frac {a}{(b-1)^2}, \frac {a+b-1}{a}}$ is not freely self-decomposable. Moreover, by Propositions 3.8 and 4.5, the measure $f\beta '(a,b)$ is unimodal if and only if

$$ \begin{align*}1 < \frac{a+b-1}{a} \le 1+ \frac{\frac{a}{b-1}}{\frac{a}{(b-1)^2}} = b, \end{align*} $$

which is equivalent to $a\ge 1$ .

5.1 Gibbs measure associated with potential

We study the Gibbs measure associated with potential $V_{t,\theta ,\lambda }$ :

$$ \begin{align*}\rho_{t,\theta,\lambda}(\mathrm{d}x) = \frac{1}{\mathcal{Z}_{t,\theta,\lambda}} \exp\{ -V_{t,\theta,\lambda}(x)\} \mathrm{ d}x \quad \text{for} \quad t,\theta>0, \ \lambda\ge1, \end{align*} $$

where $\mathcal {Z}_{t,\theta ,\lambda }$ is the normalized constant (i.e., the partition function). We first present an explicit formula for the partition function $\mathcal {Z}_{t,\theta ,\lambda }$ as follows.

Lemma 5.1 Let us consider $t,\theta>0$ and $\lambda \ge 1$ . Then,

$$ \begin{align*} \mathcal{Z}_{t,\theta,\lambda}= \begin{cases} \left(\dfrac{t^2}{\theta}\right)^{-1-\frac{t}{\theta}} \Gamma\left(1+\dfrac{t}{\theta}\right), & \lambda=1\\ (t(\lambda+1))^{-\frac{t}{\theta}-1} B\left(\dfrac{t}{\theta(\lambda-1)}, 1+\dfrac{t}{\theta}\right), & \lambda>1. \end{cases} \end{align*} $$

Proof A simple computation leads to the desired results. Actually, if $\lambda =1$ , then

$$ \begin{align*} {\mathcal{Z}}_{t,\theta,\lambda} &= \int_0^\infty x^{-(2+\frac{t}{\theta})} e^{-\frac{t^2}{\theta x}}\mathrm{d}x\\ &= \left(\frac{t^2}{\theta}\right)^{-1-\frac{t}{\theta}} \int_0^\infty u^{\frac{t}{\theta}}e^{-u}\mathrm{d}u \qquad \text{(by putting } u= t^2/(\theta x))\\ &=\left(\frac{t^2}{\theta}\right)^{-1-\frac{t}{\theta}} \Gamma \left(1+\frac{t}{\theta}\right). \end{align*} $$

If $\lambda>1$ , then

$$ \begin{align*} {\mathcal{Z}}_{t,\theta,\lambda} &= \int_0^\infty x^{-1 +\frac{t}{\theta(\lambda-1)}} \left(x +t(\lambda-1)\right)^{-1-\frac{t\lambda}{\theta(\lambda-1)}}\mathrm{d}x\\ &=\int_0^\infty u^{\frac{t}{\theta}} \left(1+t(\lambda-1)u\right)^{-1- \frac{t\lambda}{\theta(\lambda-1)}} \mathrm{d}u \qquad {(x=1/u)}\\ &=\left(t(\lambda+1)\right)^{-1-\frac{t}{\theta}} \int_0^\infty v^{\frac{t}{\theta}} (1+v)^{-1- \frac{t\lambda}{\theta(\lambda-1)}} \mathrm{d}v \qquad {(v=t(\lambda-1)u)}\\ &=\left(t(\lambda+1)\right)^{-1-\frac{t}{\theta}} B\left(1+\frac{t}{\theta}, \frac{t}{\theta(\lambda-1)}\right). \end{align*} $$

By symmetry of beta functions, we obtain the desired result.

By Lemma 5.1, the Gibbs measure $\rho _{t,\theta ,\lambda }$ has the following form:

(5.1) $$ \begin{align} \rho_{t,\theta,1} (\mathrm{d} x) &=\frac{(\frac{t^2}{\theta})^{1+\frac{t}{\theta}}}{\Gamma(1+\frac{t}{\theta})} x^{-(2+ \frac{t}{\theta})} e^{-\frac{t^2}{\theta x}} \mathrm{d}x, \end{align} $$

(5.2) $$ \begin{align} \rho_{t,\theta,\lambda}(\mathrm{d} x) &=\frac{(t(\lambda-1))^{\frac{t}{\theta}+1}}{B(\frac{t}{\theta(\lambda-1)},1+\frac{t}{\theta})} x^{-1 + \frac{t}{\theta(\lambda-1)}} \left(x+t(\lambda-1)\right)^{-1-\frac{t\lambda}{\theta(\lambda-1)}}\mathrm{d}x, \quad \lambda>1. \end{align} $$

From the above representation (5.1), the measure $\rho _{t,\theta ,1}$ is the inverse gamma distribution. More precisely,

(5.3) $$ \begin{align} \rho_{t,\theta,1} = \left(\gamma_{1+\frac{t}{\theta},\frac{\theta}{t^2}}\right)^{\langle-1\rangle}, \end{align} $$

where $(\gamma _{a,b})^{\langle -1\rangle }$ is defined by

$$ \begin{align*}(\gamma_{a,b})^{\langle -1\rangle} = \frac{1}{b^a \Gamma(a)} x^{-a-1} e^{-\frac{1}{bx}} \mathrm{d} x, \qquad a,b>0. \end{align*} $$

It is known that the class $\{(\gamma _{a,b})^{\langle -1\rangle }:a,b>0\}$ includes the positive $1/2$ -classical stable law $\sqrt {\frac {c}{2\pi }} x^{-\frac {3}{2}} e^{-\frac {c}{2x}}\mathrm {d}x$ , $c>0$ (it is also called Lévy distribution), but the measure $\rho _{t,\theta ,1}$ cannot be the positive $1/2$ -classical stable law for any $t,\theta>0$ .

For $\lambda>1$ , the formula (5.2) implies that

(5.4) $$ \begin{align} \rho_{t,\theta,\lambda} &= D_{t(\lambda-1)}\left( \beta' \left(\frac{t}{\theta(\lambda-1)}, 1+\frac{t}{\theta} \right)\right) \end{align} $$

(5.5) $$ \begin{align} &=D_{t(\lambda-1)} \left( \gamma_{\frac{t}{\theta(\lambda-1)},1} \circledast (\gamma_{1+\frac{t}{\theta},1})^{\langle -1\rangle}\right), \end{align} $$

where $\beta '(a,b)$ is the beta prime distribution, that is,

$$ \begin{align*}\beta'(a,b) := \gamma_{a,1}\circledast (\gamma_{b,1})^{\langle-1\rangle} = \frac{1}{B(a,b)} x^{-1+a} (1+x)^{-a-b} \mathrm{d} x, \qquad a,b>0. \end{align*} $$

Equation (5.5) is obtained by replacing $\boxtimes $ , $\pi _{a,1,}$ and $\mu _{t,\theta ,\lambda }$ in equation (4.1) of Theorem 4.3 with $\circledast $ , $\gamma _{a,1,}$ and $\rho _{t,\theta ,\lambda }$ , respectively. Using $\gamma _{a,b} = D_b(\gamma _{a,1})$ for $a,b>0$ , and equations (5.3) and (5.5), we get

$$ \begin{align*} \rho_{t,\theta,\lambda} = \rho_{t,\theta,1} \circledast \gamma_{\frac{t}{\theta(\lambda-1)}, \frac{\theta(\lambda-1)}{t}}. \end{align*} $$

This formula can be interpreted as the classical counterpart of equation (4.2). In particular, if we put $\lambda =1+t/\theta $ , then

$$ \begin{align*}\rho_{t,\theta,1+t/\theta}=\rho_{t,\theta,1} \circledast \gamma_{1,1}. \end{align*} $$

The measure $\rho _{t,\theta , 1+t/\theta }$ belongs to the ME (mixture of exponential distributions), that is, the distribution of the form $EZ$ , where $E,Z$ are independent random variables such that E is distributed as some exponential distribution and $Z\ge 0$ (see [Reference GoldieGol67, Reference SteutelSte67] for details). According to [Reference BondessonBon92, Reference Ismail and KelkerIK79], the inverse gamma distributions and the beta prime distributions are classically self-decomposable. Thus, for any $t,\theta>0$ and $\lambda \ge 1$ , the measure $\rho _{t,\theta ,\lambda }$ is self-decomposable.

Summarizing the discussion so far, we arrive at the following theorem.

Theorem 5.2 (Convolution formula for $\rho _{t,\theta ,\lambda }$ )

Let us consider $t,\theta>0$ , $\lambda> 1$ and ${q=\frac {t}{\theta (\lambda -1)}}$ . Then, we obtain

$$ \begin{align*} \rho_{t,\theta, \lambda} &= D_{t(\lambda-1)} \left( \gamma_{q,1} \circledast (\gamma_{1+\frac{t}{\theta},1})^{\langle -1\rangle}\right) =\rho_{t,\theta,1} \circledast \gamma_{q,q^{-1}}, \end{align*} $$

and $\rho _{t,\theta ,\lambda }$ is self-decomposable. In particular, $\rho _{t,\theta , 1+t/\theta }= \rho _{t,\theta ,1} \circledast \gamma _{1,1}$ , and hence it belongs to the ME.

Remark 5.3 Let $p_{t,\theta , \lambda }$ be the density function of the measure $\rho _{t,\theta ,\lambda }$ . We obtain the following ordinary differential equations:

$$ \begin{align*} \frac{p_{t,\theta,1}'(x)}{p_{t,\theta,1}(x)}+ \frac{x-\frac{t^2}{2\theta+t}}{\frac{\theta}{2\theta+t}x^2}=0 \end{align*} $$

and

$$ \begin{align*} \frac{p_{t,\theta,\lambda}'(x)}{p_{t,\theta,\lambda}(x)} + \frac{(x+ \frac{t(\lambda-1)}{2}) - \frac{t^2(\lambda+1)}{2(2\theta+t)}}{\frac{\theta}{2\theta+t} (x+ \frac{t(\lambda-1)}{2})^2 - \frac{t^2\theta(\lambda-1)^2}{4(2\theta+t)}} = 0, \qquad \lambda>1. \end{align*} $$

Thus, we see that the family $\{\rho _{t,\theta ,\lambda }: t,\theta>0, \lambda \ge 1\} \subset {\mathcal {P}}(\mathbb {R}_{\ge 0})$ coincides with a subfamily of Pearson distributions (see [Reference PearsonP1895, p. 381]).

5.2 Equilibrium measure of free entropy

In this section, we investigate the maximizer of free entropy associated with the potential $V_{t,\theta ,\lambda }$ .

Theorem 5.4 (Maximizer of free entropy)

For each $t,\theta>0$ and $1 \le \lambda <1+t/\theta $ , the measure $\mu _{t,\theta ,\lambda }$ is a unique maximizer of

$$ \begin{align*}\Sigma_{V_{t,\theta,\lambda}(}\mu):= \iint_{\mathbb{R}_{>0}\times \mathbb{R}_{>0}} \log |x-y|\mu(\mathrm{d} x) \mu(\mathrm{d} y) -\int_{\mathbb{R}_{>0}} V_{t,\theta,\lambda}(x)\mu(\mathrm{d} x), \end{align*} $$

among all $\mu \in {\mathcal {P}}(\mathbb {R}_{>0})$ .

Proof Thanks to the theory of the energy problem in [Reference JohanssonJoh98], the existence and uniqueness for the maximizer $\mu _V$ of $\Sigma _{V_{t,\theta ,\lambda }}$ are guaranteed. The rest of the proof is to conclude that $\mu _V=\mu _{t,\theta ,\lambda }$ . Since $V_{t,\theta ,\lambda }$ is regular enough (see [Reference Féral, Donati-Martin, Émery, Rouault and StrickerFer08, Section 4.2] and [Reference FèralFer06, Section 2]), $\mu _V$ has the density $\Phi _V=\frac {\mathrm {d}\mu _V}{\mathrm {d} x}$ and is compactly supported on $\mathbb {R}_{>0}$ , denoted by $[a,b]$ for some $0<a<b$ . We divide two cases for $\lambda $ as follows.

Case of $\boldsymbol {\lambda =1}$ : By definition of $V_{t,\theta ,1}$ , we can apply [Reference FèralFer06, Theorem 1] in the case of $\lambda =-1-t/\theta $ , $\alpha =0,$ and $\beta =t^2/\theta $ to the points $a,b$ and the density $\Phi _V$ . Then, $0<a<b$ satisfy

$$ \begin{align*}2+\frac{t}{\theta}-\frac{t^2}{\theta}\cdot \frac{a+b}{2ab}=0 \quad \text{and} \quad \sqrt{ab}=t. \end{align*} $$

Thus, we have

$$ \begin{align*} a= 2\theta+t -2\sqrt{\theta(\theta+t)} = \alpha^- \quad \text{and} \quad b=2\theta+t + 2\sqrt{\theta(\theta+t)} = \alpha^+, \end{align*} $$

where $\alpha ^\pm $ is defined in (3.2) for $\lambda =1$ . Moreover,

$$ \begin{align*} \Phi_V(x) &=\frac{1}{2\pi} \sqrt{(x-a)(b-x)} \cdot \frac{\beta}{\sqrt{ab}x^2} \mathbf{1}_{[a,b]}(x)\\ &= \frac{t\sqrt{(x-\alpha^-)(\alpha^+-x)} }{2\pi \theta x^2} \mathbf{1}_{[\alpha^-,\alpha^+]}(x) = \frac{\mathrm{d}\mu_{t,\theta,1}}{\mathrm{d} x}(x) , \end{align*} $$

as desired.

Case of $\boldsymbol {1<\lambda < 1+t/\theta }$ : By [Reference Saff and TotikST97, Theorem IV. 1.11], the points a and b satisfy the following singular integral equations:

(5.6) $$ \begin{align} \frac{1}{\pi} \int_a^b \frac{V_{t,\theta,\lambda}'(x)}{\sqrt{(b-x)(x-a)}} \mathrm{d} x=0 \quad \text{and} \quad \frac{1}{\pi} \int_a^b \frac{xV_{t,\theta,\lambda}'(x)}{\sqrt{(b-x)(x-a)}} \mathrm{d} x=2. \end{align} $$

By using the formulas

$$ \begin{align*}\int_a^b \frac{1}{x \sqrt{(b-x)(x-a)} }\mathrm{d} x = \frac{\pi}{\sqrt{ab}}\quad \text{and} \quad \int_a^b \frac{1}{\sqrt{(b-x)(x-a)} }\mathrm{d} x =\pi, \end{align*} $$

we obtain

$$ \begin{align*} \frac{1}{\pi} &\int_a^b \frac{V_{t,\theta,\lambda}'(x)}{\sqrt{(b-x)(x-a)}} \mathrm{d} x \\ &= \frac{1}{\pi} \int_a^b \frac{1-\frac{t}{\theta(\lambda-1)}}{x \sqrt{(b-x)(x-a)}}\mathrm{d} x + \frac{1}{\pi} \int_a^b \frac{1+\frac{t\lambda}{\theta(\lambda-1)}}{(x+t(\lambda-1) )\sqrt{(b-x)(x-a)}}\mathrm{d} x \\ &= \left(1-\frac{t}{\theta(\lambda-1)} \right) \frac{1}{\sqrt{ab}} + \left(1+\frac{t\lambda}{\theta(\lambda-1)}\right) \frac{1}{\sqrt{(a+t(\lambda-1))(b+t(\lambda-1))}}. \end{align*} $$

Moreover, we have

$$ \begin{align*} \frac{1}{\pi} & \int_a^b \frac{xV_{t,\theta,\lambda}'(x)}{\sqrt{(b-x)(x-a)}} \mathrm{d} x\\ &= \frac{1}{\pi} \int_a^b \frac{1-\frac{t}{\theta(\lambda-1)}}{\sqrt{(b-x)(x-a)}}\mathrm{d} x + \frac{1}{\pi} \int_a^b \frac{\left(1+\frac{t\lambda}{\theta(\lambda-1)}\right)x}{(x+t(\lambda-1) )\sqrt{(b-x)(x-a)}}\mathrm{d} x \\ &= \left(1-\frac{t}{\theta(\lambda-1)}\right) +\left(1+\frac{t\lambda}{\theta(\lambda-1)}\right)\left\{1- \frac{t(\lambda-1)}{\sqrt{(a+t(\lambda-1))(b+t(\lambda-1))}}\right\}\\ &=2+\frac{t}{\theta} -\left(1+\frac{t\lambda}{\theta(\lambda-1)}\right)\frac{t(\lambda-1)}{\sqrt{(a+t(\lambda-1))(b+t(\lambda-1))}}. \end{align*} $$

Therefore, the equations (5.6) imply that

(5.7) $$ \begin{align} \begin{cases} \left(1+\dfrac{t\lambda}{\theta(\lambda-1)}\right) \dfrac{1}{\sqrt{(a+t(\lambda-1))(b+t(\lambda-1))}} = \left(\dfrac{t}{\theta(\lambda-1)} -1\right) \dfrac{1}{\sqrt{ab}} \\ \left(1+\dfrac{t\lambda}{\theta(\lambda-1)}\right)\dfrac{1}{\sqrt{(a+t(\lambda-1))(b+t(\lambda-1))}} = \dfrac{1}{\theta(\lambda-1)}. \end{cases} \end{align} $$

By solving the above equations, we have

(5.8) $$ \begin{align} a+b = 2(\theta(\lambda+1)+t)\quad \text{and} \quad ab = (\theta(\lambda-1)-t)^2. \end{align} $$

This implies that $a,b$ are the solution of $(z-\alpha ^+)(z-\alpha ^-)=0$ , and hence $a= \alpha ^-$ and $b=\alpha ^+$ .

By using the formula

$$ \begin{align*}\text{p.v}\left(\frac{1}{\pi} \int_a^b \frac{1}{u\sqrt{(u-a)(b-u)}} \frac{{\mathrm{d} u}}{u-x} \right) =-\frac{1}{\sqrt{ab}x}, \end{align*} $$

we get

$$ \begin{align*} \text{p.v.}& \left(\frac{1}{\pi}\int_{\alpha^-}^{\alpha^+} \frac{V_{t,\theta,\lambda}'(u)}{\sqrt{(u-\alpha^-)(\alpha^+-u)}} \frac{\mathrm{ d} u}{u-x}\right)\\ &=\left(1-\frac{t}{\theta(\lambda-1)}\right)\text{p.v.} \left(\frac{1}{\pi}\int_{\alpha^-}^{\alpha^+} \frac{1}{u\sqrt{(u-\alpha^-)(\alpha^+-u)}} \frac{\mathrm{d} u}{u-x}\right)\\ &\hspace{6mm}+\left(1+\frac{t\lambda}{\theta(\lambda-1)}\right) \text{p.v.} \left(\frac{1}{\pi}\int_{\alpha^-}^{\alpha^+} \frac{1}{(u+t(\lambda-1))\sqrt{(u-\alpha^-)(\alpha^+-u)}} \frac{\mathrm{d} u}{u-x}\right)\\ &=-\left(1-\frac{t}{\theta(\lambda-1)}\right)\frac{1}{\sqrt{\alpha^+\alpha^-} x}\\ &\hspace{6mm} -\left(1+\frac{t\lambda}{\theta(\lambda-1)}\right) \frac{1}{\sqrt{(\alpha^-+t(\lambda-1))(\alpha^++t(\lambda-1))}}\cdot \frac{1}{x+t(\lambda-1)}\\ &=\frac{1}{\theta(\lambda-1)x} -\frac{1}{\theta(\lambda-1)}\cdot\frac{1}{x+t(\lambda-1)} \qquad \text{(by }({5.7})\text{ and }({5.8}))\\ &=\frac{t}{\theta x (x+t(\lambda-1))}. \end{align*} $$

From [Reference Saff and TotikST97, Theorem IV. 3.1], we finally obtain

$$ \begin{align*} \Phi_V(x)&=\frac{1}{2\pi} \sqrt{(x-\alpha^-)(\alpha^+ -x)} \times \text{p.v.} \left( \frac{1}{\pi} \int_{\alpha^-}^{\alpha^+} \frac{V_{t,\theta,\lambda}'(u)}{\sqrt{(u-\alpha^-)(\alpha^+-u)}} \frac{\mathrm{d} u}{u-x}\right)\\ &= \frac{t\sqrt{(x-\alpha^-)(\alpha^+ -x)} }{2\pi \theta x (x+t(\lambda-1))} = \frac{\mathrm{d} \mu_{t,\theta,\lambda}}{\mathrm{d} x}(x), \end{align*} $$

as desired.

Due to Theorem 5.4, the potential correspondence maps the measure $\rho _{t,\theta ,\lambda }$ to the measure $\mu _{t,\theta ,\lambda }$ for all $t,\theta>0$ and $1\le \lambda <1+t/\theta $ . In particular, the potential correspondence maps the beta prime distributions $\beta '(a,b)$ to the free ones $f\beta '(a,b)$ for all $a,b>1$ , due to Proposition 4.5 and (5.4). Consequently, we obtain the following result for free beta prime distributions.

Corollary 5.6 Let us consider $a,b>1$ . The measure $f\beta '(a,b)$ is the unique maximizer of the free entropy $\Sigma _{V_{a,b}}(\mu )$ among probability measures $\mu $ on $\mathbb {R}_{>0}$ , where

$$ \begin{align*}V_{a,b}(x) = (1-a) \log x + (a+b) \log(1+x), \qquad x>0. \end{align*} $$

7.1 Jacobi polynomial

For $a \in \mathbb {R}\setminus \{i/d: i=1,\dots , d-1\}$ and $b\in \mathbb {R}$ , we denote by $J^{(a,b)}_d$ the monic polynomial of degree d, in which its coefficient is given by

$$ \begin{align*}\widetilde{e}_k^{(d)}\left(J^{(a,b)}_d \right) := \frac{(bd)_k}{(ad)_k} \qquad \text{for} \ k=0, 1,\dots,d, \end{align*} $$

where $(x)_n:=x(x-1)\dots (x-n+1)$ and $(x)_0:=1$ . The polynomials $J^{(a,b)}_d$ are well-known as the Jacobi polynomials. According to [Reference Martínez-Finkelshtein, Morales and PeralesMFMP24b, (80)], the polynomial $J^{(a,b)}_d$ can be represented by a hypergeometric function as follows:

$$ \begin{align*}J_d^{(a,b)}(x)=(-1)^d\frac{ (bd)_d}{(ad)_d} {}_2F_1 (-d, ad-d+1; bd-d+1; x). \end{align*} $$

It is known that the polynomial is orthogonal with respect to the weight function

$$ \begin{align*}W_{d}^{(a,b)}(x):=x^{d(b-1)}(1-x)^{d(a-b-1)} \end{align*} $$

when $-bd+d-1 \notin \mathbb {Z}_{\ge 0}$ (see, e.g., [Reference Dominici, Johnston and JordaanDJJ13, Theorem 1]). Recently, the class of hypergeometric polynomials (including $J^{(a,b)}_d$ ) was studied in the framework of finite free probability theory (see, e.g., [Reference Martínez-Finkelshtein, Morales and PeralesMFMP24a, Reference Martínez-Finkelshtein, Morales and PeralesMFMP24b, Reference Arizmendi, Fujie, Perales and UedaAFPU24]).

In particular, we define

$$ \begin{align*}\widehat{J}^{(a,b)}_d(x):= J^{(a,b)}_d(-x). \end{align*} $$

By [Reference Martínez-Finkelshtein, Morales and PeralesMFMP24b, Section 5.2] and [Reference Dominici, Johnston and JordaanDJJ13, Proposition 4 and Theorem 5], the polynomial $\widehat {J}^{(a,b)}_d$ has d distinct roots in which are all nonnegative when $a<0$ and $b>1$ . Furthermore, we define the monic real-rooted polynomial of degree d by

$$ \begin{align*}p_d^{(t,\theta,\lambda)}:= D_{t(\lambda-1)} (\widehat{J}^{\ (A, B) }_d), \qquad \text{where} \quad A=-\frac{t}{\theta} \quad \text{and} \quad B=\frac{t}{\theta(\lambda-1)} +\frac{1}{d}, \end{align*} $$

for $t,\theta>0$ and $1<\lambda \le 1+t/\theta $ . In this case, it is easy to check $A<0$ and $B>1$ , and therefore $p_d^{(t,\theta ,\lambda )}$ also has d distinct roots in which are all nonnegative. Moreover, $p_d^{(t,\theta ,\lambda )}$ is orthogonal with respect to the weight function

(7.1) $$ \begin{align} W_d^{(A,B)}\left(-\frac{x}{t(\lambda-1)}\right) \propto x^{d(B-1)}(x+t(\lambda-1))^{d(A-B-1)}. \end{align} $$

According to recent work [Reference Arizmendi, Fujie, Perales and UedaAFPU24], the ratio of consecutive coefficients of a polynomial plays a role in the S-transform in the framework of finite free probability. In what follows, we apply the results of [Reference Arizmendi, Fujie, Perales and UedaAFPU24] to the sequence of polynomials $(p_d^{(t,\theta ,\lambda )})_{d\in \mathbb {N}}$ . A direct computation shows that

$$ \begin{align*}\frac{\widetilde{e}_{k-1}^{(d)}(p_d^{(t,\theta,\lambda)})}{\widetilde{e}_{k}^{(d)}(p_d^{(t,\theta,\lambda)})} = \frac{1}{t(\lambda-1)} \frac{\frac{t}{\theta}+\frac{k-1}{d}}{\frac{t}{\theta(\lambda-1)}-\frac{k-2}{d}}. \end{align*} $$

As $d\to \infty $ with $k/d\to z\in (0, 1)$ , the limiting value of the above consecutive coefficient ratio can be computed as follows:

$$ \begin{align*} \frac{\widetilde{e}_{k-1}^{(d)}(p_d^{(t,\theta,\lambda)})}{\widetilde{e}_{k}^{(d)}(p_d^{(t,\theta,\lambda)})} &\to \frac{1}{t(\lambda-1)} \frac{\frac{t}{\theta}+z}{\frac{t}{\theta(\lambda-1)}-z} \\ &= \frac{1}{t(\lambda-1)} \frac{(1+\frac{t}{\theta}) -1 - (-z)}{\frac{t}{\theta(\lambda-1)} +(-z)} \\ &= S_{D_{t(\lambda-1)} (f\beta'(\frac{t}{\theta(\lambda-1)}, 1+\frac{t}{\theta}))}(-z) \qquad \text{(by }({2.2})\text{ and }({4.4}))\\ &= S_{\mu_{t,\theta,\lambda}}(-z) \qquad \text{(by }{4.3}). \end{align*} $$

By [Reference Arizmendi, Fujie, Perales and UedaAFPU24, Theorem 1.1], the limiting behavior of the empirical root measure of $ p_d^{(t,\theta ,\lambda )}$ can be described as follows.

Proposition 7.1 For $t,\theta>0$ and $1<\lambda \le 1+t/\theta $ , we have

$$ \begin{align*}\mathfrak{m}[[p_d^{(t,\theta,\lambda)}]] \xrightarrow{w} \mu_{t,\theta,\lambda} \qquad \text{as} \qquad d\to\infty. \end{align*} $$

7.2 Bessel polynomial

For $a\in \mathbb {R} \setminus \{i/d: i=1, \dots , d-1\}$ , we define $B^{(a)}_d$ as the monic polynomial of degree d, whose coefficients are given by

$$ \begin{align*}\widetilde{e}_k^{(d)}\left(B^{(a)}_d \right) = \frac{d^k}{(a d)_k} \qquad \text{for} \ k=0,1,\dots, d. \end{align*} $$

The polynomials $B^{(a)}_d$ are well-known as the Bessel polynomials. According to [Reference Martínez-Finkelshtein, Morales and PeralesMFMP24b, Equation (80)], the polynomial $B^{(a)}_d$ can be represented by a hypergeometric polynomial as follows:

$$ \begin{align*}B^{(a)}_d(x) = \frac{(-1)^d}{(ad)_d} {}_2 F_0 (-d, a d-d+1; - \; x). \end{align*} $$

We define

$$ \begin{align*}\widehat{B}_d^{(a)}(x):= B_d^{(a)}(-x). \end{align*} $$

According to [Reference Martínez-Finkelshtein, Morales and PeralesMFMP24b], $\widehat {B}_d^{(a)}$ has d distinct roots in which are all nonnegative whenever $a<0$ . Furthermore, we define the monic real-rooted polynomial of degree d by

$$ \begin{align*}p_d^{(t,\theta,1)} : = D_{t^2/\theta}\left(\widehat{B}_d^{(-t/\theta)}\right), \qquad t,\theta>0. \end{align*} $$

Since $-t/\theta <0$ , the polynomial $p_d^{(t,\theta ,1)}$ also has d distinct roots in which are all nonnegative. Then, we obtain

$$ \begin{align*} \frac{\widetilde{e}_{k-1}^{(d)}(p_d^{(t,\theta,1)})}{\widetilde{e}_{k}^{(d)}(p_d^{(t,\theta,1)})} = \frac{1}{t^2}\left(t +\theta\cdot \frac{k-1}{d}\right) \to \frac{1}{t^2} (t+\theta z) = S_{\mu_{t,\theta,1}}(-z) \end{align*} $$

as $k/d\to z \in (0,1)$ . According to [Reference Arizmendi, Fujie, Perales and UedaAFPU24], we obtain the following result.

Proposition 7.2 For $t,\theta>0$ , we get

$$ \begin{align*}\mathfrak{m}[[p_d^{(t,\theta,1)}]] \xrightarrow{w} \mu_{t,\theta,1} \qquad \text{as} \qquad d\to \infty. \end{align*} $$

7.3 Finite free version of convolution formula

In this section, we construct the finite analog of Theorem 4.3. To begin, we introduce a finite version of the free multiplicative convolution. For monic polynomials $p,q$ of degree d, their finite free multiplicative convolution $p\boxtimes _d q$ is defined as the monic polynomial whose coefficients satisfy

$$ \begin{align*}\widetilde{e}_k^{(d)} (p\boxtimes_d q) := \widetilde{e}_k^{(d)}(p)\widetilde{e}_k^{(d)}(q), \qquad k=1,\dots, d. \end{align*} $$

It is known that if $p,q$ are nonnegative real rooted, then so is $p\boxtimes _d q$ . For the relationship between $\boxtimes $ and $\boxtimes _d$ , see [Reference Arizmendi, Garza-Vargas and PeralesAGP23, Reference Marcus, Spielman and SrivastavaMSS22].

Given $\lambda>0$ and $\theta>0$ , define the Laguerre polynomial $L_d^{(\lambda ,\theta )}$ to be the monic polynomial whose coefficients are

$$ \begin{align*}\widetilde{e}_k^{(d)}\left(L^{(\lambda,\theta)}_d \right) := \theta^k \frac{(\lambda d)_k}{d^k}, \qquad k=1,\dots, d. \end{align*} $$

The polynomial $L_d^{(\lambda ,\theta )}$ has d distinct, strictly positive roots whenever $\lambda>0$ . Moreover, it is known that $\mathfrak {m}[[L_d^{(\lambda ,\theta )}]]\xrightarrow {w} \pi _{\lambda ,\theta }$ as $d\to \infty $ .

Combining the above observations, we arrive at the following formula.

Proposition 7.3 For $t, \theta>0$ , $\lambda>1,$ and $d\in \mathbb {N}$ , we have

$$ \begin{align*}p_d^{(t,\theta,\lambda)} = p_d^{(t,\theta,1)} \boxtimes_d L_d^{(q+ 1/d,\ q^{-1})}, \end{align*} $$

where $q= \frac {t}{\theta (\lambda -1)}$ .

Proof We compare the k-th coefficients of $p_d^{(t,\theta ,\lambda )}$ and $ p_d^{(t,\theta ,1)} \boxtimes _d L_d^{(q+ 1/d,\ q^{-1})}$ . First, we have

$$ \begin{align*} \widetilde{e}_k^{(d)}(p_d^{(t,\theta,\lambda)}) &= t^k (\lambda-1)^k (-1)^{d-k}\ \widetilde{e}_k^{(d)} (J_d^{(-t/\theta,\ q+ 1/d)})\\ &= t^k (\lambda-1)^k (-1)^{d-k}\ \frac{(qd+1)_k}{(-\frac{t}{\theta}d)_k}. \end{align*} $$

On the other hand, we get

$$ \begin{align*} \widetilde{e}_k^{(d)}( p_d^{(t,\theta,1)} \boxtimes_d L_d^{(q+ 1/d,\ q^{-1})}) &=\widetilde{e}_k^{(d)}(p_d^{(t,\theta,1)})\widetilde{e}_k^{(d)}(L_d^{(q+ 1/d,\ q^{-1})})\\ &= \frac{t^{2k}}{\theta^k} (-1)^{d-k} \frac{d^k}{(-\frac{t}{\theta}d)_k} \times \frac{\theta^k(\lambda-1)^k}{t^k} \frac{(qd+1)_k}{d^k}\\ &= t^k (\lambda-1)^k (-1)^{d-k}\ \frac{(qd+1)_k}{(-\frac{t}{\theta}d)_k}, \end{align*} $$

as desired.

7.4 Comments

These results (Propositions 7.1 and 7.2) are not essentially new since Martínez-Finkelshtein et al. [Reference Martínez-Finkelshtein, Morales and PeralesMFMP24a, Reference Martínez-Finkelshtein, Morales and PeralesMFMP24b] and Arizmendi et al. [Reference Arizmendi, Fujie, Perales and UedaAFPU24] have already studied the relationship between the asymptotic behavior of the empirical root measures of Jacobi or Bessel polynomials and free probability. Nevertheless, we would like to emphasize that our analysis provides deeper insights into the behavior of the roots of Jacobi or Bessel polynomials, thanks to the understanding gained from the distributional properties of the Meixner-type free gamma distribution and its connection with free entropy. For instance, the weight function (7.1) of $p_d^{(t,\theta ,\lambda )}$ belongs to the Pearson class, which may be connected to the Gibbs measure $\rho _{t,\theta ,\lambda }$ associated with the potential $V_{t,\theta ,\lambda }$ discussed in Section 5.