On Generalized V_a-Transformation of Measures

Abstract

In this study, we introduce a novel transformation of probability measures that unifies two significant transformations in free probability theory: the t-transformation and the $V_a$-transformation. Our unified transformation, denoted ${\mathcal{U}}_{(a,t)}$, is defined analytically via a modified functional equation involving the Cauchy transform, and reduces to the t-transformation when $a=0$, and to the $V_a$-transformation when $t=1$. We investigate some properties of this new transformation from the lens of Cauchy–Stieltjes kernel (CSK) families and the corresponding variance functions (VFs). We derive a general expression for the VF resulting from the ${\mathcal{U}}_{(a,t)}$-transformation. This new expression is applied to prove a central result: the free Meixner family (FMF) of measures is invariant under this transformation. Furthermore, novel limiting theorems involving ${\mathcal{U}}_{(a,t)}$-transformation are proved providing new insights into the relations between some important measures in free probability such as the semicircle, Marchenko–Pastur, and free binomial measures.

1. Introduction

In free probability, the investigation of measure transformations has proven to be a fruitful approach for understanding the interplay between convolution operations and the analytic structures they induce. Two such transformations that have attracted particular attention are the t-transformation [1,2], which provides a deformation mechanism linking classical and free convolution-type operations, and the $V_a$-transformation [3], which describes structural modifications of measures preserving key properties of free convolution semigroups. Each of these maps encapsulates distinct aspects of stability, divisibility, and parameter dependence in non-commutative probability, and both have become essential tools in the analysis of Cauchy–Stieltjes kernel (CSK) families, and limit theorems. Despite their individual significance, the t- and $V_a$-transformations have so far been studied separately, leaving open the possibility of a unified framework that can accommodate their common features and reveal deeper structural insights. The motivation for introducing such a unified transformation lies in its potential to provide a more general analytical tool for studying the evolution and deformation of measures under free, Boolean, and classical convolutions. A unified approach also offers the opportunity to obtain new characterizations of fundamental distributions in free probability, and to develop a systematic method for exploring stability and infinite divisibility across different convolution structures.

From a broader perspective, the results presented here contribute to linking abstract concepts from free probability with ideas that are familiar to statisticians and geometers. In particular, the structure of CSK families and their variance functions (VFs) can be viewed as non-commutative analogues of exponential families and Fisher information metrics in classical statistics. This connection highlights how deformation processes in free probability can be interpreted geometrically, offering potential applications in areas such as quantum statistics, information geometry, and complex system modeling. Thus, the ${\mathcal{U}}_{(a,t)}$-framework not only extends the theoretical boundaries of non-commutative probability but also opens the door to cross-disciplinary applications where probabilistic, geometric, and statistical ideas intersect. By providing explicit analytic tools and invariance results, this study contributes to a deeper understanding of the mathematical structures that underlie both random matrix phenomena and the geometric aspects of statistical inference.

Let $\mathbb{P}$ (and ${\mathbb{P}}_c$) denote the set of real probabilities (and with compact support). The idea of the t-transformation of measures uses the Cauchy–Stieltjes transform ${\mathcal{G}}_{\rho }(·)$ defined, for $\rho \in \mathbb{P}$, as

$${\mathcal{G}}_{\rho }\left(z\right)=\int {\displaystyle \frac{\rho \left(dr\right)}{z-r}},z\in \mathbb{C}\setminus supp\left(\rho \right).$$

A transformation of the Cauchy transform of $\rho \in \mathbb{P}$ is considered, in [1,2], as follows: let $t>0$; due to the Nevanlinna theorem, the new function ${\mathcal{G}}_{{\rho }_t}(·)$, introduced as

$${\displaystyle \frac{1}{{\mathcal{G}}_{{\rho }_t}\left(z\right)}}={\displaystyle \frac{t}{{\mathcal{G}}_{\rho }\left(z\right)}}+(1-t)z,$$

turns out to be the Cauchy transform of some probability measure indicated as $U_t\left(\rho \right):={\rho }_t$. The t-transformation of $\rho$ is precisely the t-th Boolean additive convolution power of $\rho$, i.e., $U_t\left(\rho \right)={\rho }^{\uplus t}$; see [4] for more details about the Boolean additive convolution ⊎.

Another transformation of the Cauchy transform for $\rho \in \mathbb{P}$ (with finite variance) is defined in [3]. The transformation adds a fixed proportion of the variance of $\rho$ to the reciprocal of the Cauchy transform of $\rho$. According to the Nevanlinna theorem, the resulting function is itself the reciprocal of the Cauchy transform of a unique probability measure. This measure is indicated by $V_a\left(\rho \right)$ and called the $V_a$-transformation of $\rho$; that is,

$${\displaystyle \frac{1}{{\mathcal{G}}_{V_a\left(\rho \right)}\left(z\right)}}={\displaystyle \frac{1}{{\mathcal{G}}_{\rho }\left(z\right)}}+a{\mathbf{r}}_2\left(\rho \right),$$

where $a\in \mathbb{R}$ and ${\mathbf{r}}_2\left(\rho \right)$ is the second free cumulant (the variance) of $\rho$.

Although these transformations have distinct origins and motivations, they share several structural features and analytic behaviors. Motivated by this observation, we propose a new transformation of measures that unifies both the t-transformation and the $V_a$-transformation into a single, more general framework. This unified transformation is designed to provide a more flexible analytical tool to study structural properties of measures under free convolution and may serve as a foundation for a broader class of transformations that preserve key invariance or stability properties in free probability. Consider $\rho \in \mathbb{P}$ with finite second moment. We introduce a new transformation of the measure for $a\in \mathbb{R}$ and $t>0$, which we denote by ${\mathcal{U}}_{(a,t)}$ as

$${\displaystyle \frac{1}{{\mathcal{G}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(z\right)}}={\displaystyle \frac{t}{{\mathcal{G}}_{\rho }\left(z\right)}}+(1-t)z+at{\mathbf{r}}_2\left(\rho \right).$$

(1)

According to the Nevanlinna theorem, ${\mathcal{G}}_{U_{(a,t)}\left(\rho \right)}(·)$ is a Cauchy transform. Note that for $a=0$, we have ${\mathcal{U}}_{(0,t)}=U_t$, and for $t=1$, we get ${\mathcal{U}}_{(a,1)}=V_a$.

The ${\mathcal{U}}_{(a,t)}$-transformation of $\rho \in {\mathbb{P}}_c$ may be illustrated by means of the continued fraction representation of the Cauchy transform. Let

$${\displaystyle {\mathcal{G}}_{\rho }\left(z\right)=\frac{1}{{\displaystyle z-{\zeta }_1-\frac{{\iota }_1}{{\displaystyle z-{\zeta }_2-\frac{{\iota }_2}{{\displaystyle z-{\zeta }_3-\frac{{\iota }_3}{z-{\zeta }_4-...}}}}}}},}$$

then,

$${\displaystyle {\mathcal{G}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(z\right)=\frac{1}{{\displaystyle z-t{\zeta }_1+at{\mathbf{r}}_2(\rho )-\frac{t{\iota }_1}{{\displaystyle z-{\zeta }_2-\frac{{\iota }_2}{{\displaystyle z-{\zeta }_3-\frac{{\iota }_3}{z-{\zeta }_4-...}}}}}}}.}$$

This article analyzes the ${\mathcal{U}}_{(a,t)}$-transformation within the framework of CSK families and their VFs. To ensure clarity, Section 2 reviews necessary concepts of CSK families. Section 3 then derives a general expression for the VF resulting from the ${\mathcal{U}}_{(a,t)}$-transformation. This new expression is applied in Section 4 to prove a central result: the free Meixner family (FMF) of measures is invariant under this transformation. Furthermore, some novel limiting theorems are established in Section 5 to demonstrate the significant power of the ${\mathcal{U}}_{(a,t)}$-transformation to generate new insights and connections within the menagerie of free probability. These results illuminate a clear and structured pathway between some cornerstone measures such as semicircle, Marchenko–Pastur, and free binomial laws.

2. CSK Families

The theory of CSK families has recently emerged as a fruitful framework within non-commutative probability, drawing strong analogies with the classical theory of natural exponential families (NEFs). In classical probability, NEFs provide a unified approach to modeling distributions through their moment-generating functions and VFs, with deep applications in statistical theory and information geometry. In the free probability context, the CSK families are formed by replacing the exponential kernel with the CSK ${(1-\vartheta x)}^{-1}$. This kernel-based construction mimics the role of the Laplace transform in NEFs, but is adapted to the analytic tools central to free probability, such as the Cauchy transform and subordination techniques. The study of CSK families thus provides a powerful new lens for exploring non-commutative analogs of classical statistical structures, with potential applications to operator theory, random matrices, and free harmonic analysis. In [5,6], the authors provide a deep analytic framework for studying CSK families, opening new directions in free statistics. This concept was developed in [7] to include probability measures with a one-sided support boundary, say from above.

${\mathbb{P}}_{ba}$ denotes the set of non-degenerate probability measures with a one-sided support boundary from above. For $\rho \in {\mathbb{P}}_{ba}$, the function

$${\mathcal{M}}_{\rho }\left(\vartheta \right)=\int {\displaystyle \frac{\rho \left(dx\right)}{1-x\vartheta }}$$

is defined for $\vartheta \in [0,{\vartheta }_{+}^{\rho })$ with $1/{\vartheta }_{+}^{\rho }=max\{0,sup\mathrm{supp}\left(\rho \right)\}$. The set

$${\mathcal{K}}_{+}\left(\rho \right)=\left\{{\mathbf{P}}_{\vartheta }^{\rho }\left(dx\right):={\displaystyle \frac{\rho \left(dx\right)}{{\mathcal{M}}_{\rho }\left(\vartheta \right)(1-\vartheta x)}};\vartheta \in (0,{\vartheta }_{+}^{\rho })\right\}$$

is called the one-sided CSK family generated by $\rho$.

According to [7], the mean function $\vartheta \mapsto {\mathbf{K}}_{\rho }\left(\vartheta \right)=\int x{\mathbf{P}}_{\vartheta }^{\rho }\left(dx\right)$ is strictly increasing on $(0,{\vartheta }_{+}^{\rho })$ and

$${\mathbf{K}}_{\rho }\left(\vartheta \right)={\displaystyle \frac{{\mathcal{M}}_{\rho }\left(\vartheta \right)-1}{\vartheta {\mathcal{M}}_{\rho }\left(\vartheta \right)}}.$$

(2)

The interval $(m_0^{\rho },m_{+}^{\rho })={\mathbf{K}}_{\rho }\left((0,{\vartheta }_{+}^{\rho })\right)$ is called the (one-sided) mean domain of ${\mathcal{K}}_{+}\left(\rho \right)$. Consider ${\psi }_{\rho }(·)$ the reciprocal of ${\mathbf{K}}_{\rho }(·)$ and for $s\in (m_0^{\rho },m_{+}^{\rho })$ write ${\mathbf{Q}}_s^{\rho }\left(dx\right)={\mathbf{P}}_{{\psi }_{\rho }\left(s\right)}^{\rho }\left(dx\right)$; then, we get the mean parametrization

$${\mathcal{K}}_{+}\left(\rho \right)=\{{\mathbf{Q}}_s^{\rho }\left(dx\right):s\in (m_0^{\rho },m_{+}^{\rho })\}.$$

With ${\mathbf{B}}_{\rho }=1/{\vartheta }_{+}^{\rho }$, it is shown in [7] that $m_0^{\rho }={lim}_{\vartheta \to 0^{+}}{\mathbf{K}}_{\rho }\left(\vartheta \right)$ and $m_{+}^{\rho }={\mathbf{B}}_{\rho }-{lim}_{z\to {\mathbf{B}}_{\rho }^{+}}1/{\mathcal{G}}_{\rho }\left(z\right)$.

If the support of $\rho$ is bounded from below, the corresponding one-sided CSK family, denoted by ${\mathcal{K}}_{-}\left(\rho \right)$, is defined. It is parameterized by $\vartheta \in ({\vartheta }_{-}^{\rho },0)$, where ${\vartheta }_{-}^{\rho }$ can be either $1/{\mathbf{A}}_{\rho }$ or $-\infty$, where ${\mathbf{A}}_{\rho }=min\{0,infsupp\left(\rho \right)\}$. For ${\mathcal{K}}_{-}\left(\rho \right)$, the mean domain is $(m_{-}^{\rho },m_0^{\rho })$ with $m_{-}^{\rho }={\mathbf{A}}_{\rho }-1/{\mathcal{G}}_{\rho }\left({\mathbf{A}}_{\rho }\right)$. If $\rho \in {\mathbb{P}}_c$, then $\vartheta \in ({\vartheta }_{-}^{\rho },{\vartheta }_{+}^{\rho })$ and the two-sided CSK family is $\mathcal{K}\left(\rho \right)={\mathcal{K}}_{+}\left(\rho \right)\cup {\mathcal{K}}_{-}\left(\rho \right)\cup \left\{\rho \right\}$.

Let $\rho \in {\mathbb{P}}_{ba}$. The VF is defined by [5]

$$s\mapsto {\mathcal{V}}_{\rho }\left(s\right)=\int {(x-s)}^2{\mathbf{Q}}_s^{\rho }\left(dx\right).$$

This is a central concept in the CSK families, playing a role analogous to that of the VF in NEFs in classical probability. It provides a simple functional relationship between the mean and variance, thereby encoding essential structural information about the family. Through the VF, one can classify CSK families, study their stability under transformations, and characterize fundamental distributions. Moreover, the VF serves as a unifying analytic tool that links moment structures, generating functions, and convolution properties, making it a powerful instrument for both theoretical developments and applications in free probability.

If $\rho \in {\mathbb{P}}_{ba}$ does not have a finite mean, in ${\mathcal{K}}_{+}\left(\rho \right)$, all the laws have infinite variance. In [7], to replace the absence of VF, a pseudo-variance function (PVF) ${\mathbb{V}}_{\rho }(·)$ notion is defined as

$${\mathbb{V}}_{\rho }\left(s\right)=s\left(1/{\psi }_{\rho }\left(s\right)-s\right).$$

(3)

If $m_0^{\rho }=\int x\rho \left(dx\right)$ is finite, the VF exists and we have [7]

$${\mathbb{V}}_{\rho }\left(s\right)={\displaystyle \frac{s}{s-m_0^{\rho }}}{\mathcal{V}}_{\rho }\left(s\right).$$

(4)

The following remark gathers all the facts that help to prove the main results of this article.

Remark 1.

Let $\rho \in {\mathbb{P}}_{ba}$.

(i) 

The PVF ${\mathbb{V}}_{\rho }(·)$ determines ρ: If we set $\omega =\omega \left(s\right)=s+{\displaystyle \frac{{\mathbb{V}}_{\rho }\left(s\right)}{s}},$ then ${\mathcal{G}}_{\rho }\left(\omega \right)={\displaystyle \frac{s}{{\mathbb{V}}_{\rho }\left(s\right)}}.$ In addition, if $m_0^{\rho }$ is finite, then

$${\mathcal{G}}_{\rho }\left(\omega \right)={\displaystyle \frac{s-m_0^{\rho }}{{\mathcal{V}}_{\rho }\left(s\right)}}.$$

(5)

Thus, the VF ${\mathcal{V}}_{\rho }(·)$ and the mean $m_0^{\rho }$ of ρ determine the generating measure ρ.

(ii) 

Let $\phi \left(\rho \right)$ be the image of ρ by $\phi :x\mapsto \alpha x+\beta$ with $\beta \in \mathbb{R}$ and $\alpha \ne 0$. Then, $m_0^{\phi \left(\rho \right)}=\phi \left(m_0^{\rho }\right)=\alpha m_0^{\rho }+\beta$ and for s close enough to $\alpha m_0^{\rho }+\beta$, the PVF is

$${\mathbb{V}}_{\phi \left(\rho \right)}\left(s\right)={\displaystyle \frac{{\alpha }^{2}s}{s-\beta }}{\mathbb{V}}_{\rho }\left({\displaystyle \frac{s-\beta }{\alpha }}\right).$$

If the VF exists, then

$${\mathcal{V}}_{\phi \left(\rho \right)}\left(s\right)={\alpha }^2{\mathcal{V}}_{\rho }\left({\displaystyle \frac{s-\beta }{\alpha }}\right).$$

(6)

(iii) 

If the variance of ρ is finite, we have ${\mathcal{U}}_{(a,t)}\left(\rho \right)=U_t\left(V_a\left(\rho \right)\right)=V_a\left(U_t\left(\rho \right)\right)$. In fact,

$${\displaystyle \frac{1}{{\mathcal{G}}_{V_a\left(U_t\left(\rho \right)\right)}\left(z\right)}}={\displaystyle \frac{1}{{\mathcal{G}}_{U_t\left(\rho \right)}\left(z\right)}}+a{\mathit{r}}_2\left(U_t\left(\rho \right)\right)={\displaystyle \frac{1}{{\mathcal{G}}_{\rho }\left(z\right)}}+(1-t)z+at{\mathit{r}}_2\left(\rho \right)={\displaystyle \frac{1}{{\mathcal{G}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(z\right)}},$$

and

$${\displaystyle \frac{1}{{\mathcal{G}}_{U_t\left(V_a\left(\rho \right)\right)}\left(z\right)}}={\displaystyle \frac{t}{{\mathcal{G}}_{V_a\left(\rho \right)}\left(z\right)}}+(1-t)z={\displaystyle \frac{t}{{\mathcal{G}}_{\rho }\left(z\right)}}+at{\mathit{r}}_2\left(\rho \right)+(1-t)z={\displaystyle \frac{1}{{\mathcal{G}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(z\right)}}.$$

Thus,

$${\mathcal{G}}_{V_a\left(U_t\left(\rho \right)\right)}\left(z\right)={\mathcal{G}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(z\right)={\mathcal{G}}_{U_t\left(V_a\left(\varrho \right)\right)}\left(z\right).$$

Furthermore, the variance of ${\mathcal{U}}_{(a,t)}\left(\rho \right)$ is ${\mathit{r}}_2\left({\mathcal{U}}_{(a,t)}\left(\rho \right)\right)=t{\mathit{r}}_2\left(\rho \right)$.

3. ${\mathcal{U}}_{(\mathit{a},\mathit{t})}$-Transformation and VF

We present some results concerning the ${\mathcal{U}}_{(a,t)}$-transformation. Consider $\rho \in {\mathcal{P}}_{ba}$ having finite variance. The result below is connected to the mean function.

Proposition 1.

The deformed measure ${\mathcal{U}}_{(a,t)}\left(\rho \right)$ belongs to ${\mathcal{P}}_{ba}$ and for $\vartheta >0$ sufficiently close to 0, one has

$${\mathit{K}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(\vartheta \right)=t{\mathit{K}}_{\rho }\left(\vartheta \right)-at{\mathit{r}}_2\left(\rho \right).$$

(7)

Proof. 

Because $\rho$ is bounded from above, it is clear from (1) that ${\mathcal{U}}_{(a,t)}\left(\rho \right)$ is similarly bounded from above. We also have

$${\mathcal{M}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(\vartheta \right)={\displaystyle \frac{{\mathcal{M}}_{\rho }\left(\vartheta \right)}{t+[(1-t)+at{\mathbf{r}}_2\left(\rho \right)\vartheta {\mathcal{M}}_{\rho }\left(\vartheta \right)]}}.$$

(8)

It is clear that ${\mathcal{M}}_{\rho }\left(0\right)=1$ and, for $\vartheta >0$ close sufficiently to 0, ${\mathcal{M}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(\vartheta \right)$ exists finitely. Relations (2) and (8) give

$${\mathbf{K}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(\vartheta \right)={\displaystyle \frac{{\mathcal{M}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(\vartheta \right)-1}{\vartheta {\mathcal{M}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(\vartheta \right)}}=t{\displaystyle \frac{{\mathcal{M}}_{\rho }\left(\vartheta \right)-1}{\vartheta {\mathcal{M}}_{\rho }\left(\vartheta \right)}}-at{\mathbf{r}}_2\left(\rho \right)=t{\mathbf{K}}_{\rho }\left(\vartheta \right)-at{\mathbf{r}}_2\left(\rho \right).$$

□

We then show how the ${\mathcal{U}}_{(a,t)}$-transformation of $\rho$ affects the VF of ${\mathcal{K}}_{+}\left(\rho \right)$.

Theorem 1.

The mean of the deformed measure is $m_0^{{\mathcal{U}}_{(a,t)}\left(\rho \right)}=t{m}_0^{\rho }-at{\mathit{r}}_2\left(\rho \right)$ and for $s>m_0^{{\mathcal{U}}_{(a,t)}\left(\rho \right)}$ close enough to $m_0^{{\mathcal{U}}_{(a,t)}\left(\rho \right)}$, one has

$${\mathbb{V}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(s\right)=s\left[{\displaystyle \frac{{\mathbb{V}}_{\rho }(\frac{s}{t}+a{\mathit{r}}_2\left(\rho \right))}{\frac{s}{t}+a{\mathit{r}}_2\left(\rho \right)}}+{\displaystyle \frac{s}{t}}+a{\mathit{r}}_2\left(\rho \right)-s\right].$$

(9)

Furthermore,

$${\mathcal{V}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(s\right)=t{\mathcal{V}}_{\rho }\left({\displaystyle \frac{s}{t}}+a{\mathit{r}}_2\left(\rho \right)\right)+\left({\displaystyle \frac{s}{t}}+a{\mathit{r}}_2\left(\rho \right)-s\right)\left(s+at{\mathit{r}}_2\left(\rho \right)-t{m}_0^{\rho }\right).$$

(10)

Proof. 

We have that

$${\displaystyle m_0^{{\mathcal{U}}_{(a,t)}\left(\rho \right)}=\underset{\vartheta ⟶0^{+}}{lim}{\mathbf{K}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}\left(\vartheta \right)=\underset{\vartheta ⟶0^{+}}{lim}t{\mathbf{K}}_{\rho }\left(\vartheta \right)-at{\mathbf{r}}_2\left(\rho \right)=t{m}_0^{\rho }-at{\mathbf{r}}_2\left(\rho \right).}$$

For $\vartheta >0$ close sufficiently to 0, put $m={\mathbf{K}}_{\rho }\left(\vartheta \right)$. From relation (7), we obtain

$${\psi }_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}(tm-at{\mathbf{r}}_2\left(\rho \right))=\vartheta ={\psi }_{\rho }\left(m\right).$$

(11)

Combining (11) with (3), we get

$${\displaystyle \frac{{\mathbb{V}}_{{\mathcal{U}}_{(a,t)}\left(\rho \right)}(tm-at{\mathbf{r}}_2\left(\rho \right))}{tm-at{\mathbf{r}}_2\left(\rho \right)}}+tm-at{\mathbf{r}}_2\left(\rho \right)={\displaystyle \frac{{\mathbb{V}}_{\rho }\left(m\right)}{m}}+m.$$

(12)

Set $s=tm-at{\mathbf{r}}_2\left(\rho \right)$; so, relation (12) is nothing but (9). Furthermore, because $\rho$ has a finite second moment, $m_0^{\rho }$ is finite, and so the VFs of ${\mathcal{K}}_{+}\left(\rho \right)$ and ${\mathcal{K}}_{+}\left({\mathcal{U}}_{(a,t)}\left(\rho \right)\right)$ exist. Relation (10) is formed by combining (9) and (4). □

Remark 2.

Note that for $t=1$, Formula (10) is reduced to the result given in ([8], Theorem 3.2, Equation (3.4)). That is, for s close enough to $m_0^{\rho }-a{\mathit{r}}_2\left(\rho \right)$,

$${\mathcal{V}}_{{\mathcal{U}}_{(a,1)}\left(\rho \right)}\left(s\right)={\mathcal{V}}_{V_a\left(\rho \right)}\left(s\right)={\mathcal{V}}_{\rho }\left(s+a{\mathit{r}}_2\left(\rho \right)\right)+a{\mathit{r}}_2\left(\rho \right)\left(s+a{\mathit{r}}_2\left(\rho \right)-m_0^{\rho }\right).$$

(13)

For $a=0$, Formula (10) is reduced to the result given in ([9], Theorem 2.3, Equation (2.19)). That is, for s close enough to $t{m}_0^{\rho }$,

$${\mathcal{V}}_{{\mathcal{U}}_{(0,t)}\left(\rho \right)}\left(s\right)={\mathcal{V}}_{U_t\left(\rho \right)}\left(s\right)={\mathcal{V}}_{{\rho }^{\uplus t}}\left(s\right)=t{\mathcal{V}}_{\rho }\left(s/t\right)+(1/t-1)s\left(s-m_0^{\rho }\right).$$

4. Invariance of the FMF Under ${\mathcal{U}}_{(\mathit{a},\mathit{t})}$-Transformation

The invariance property of the free Meixner family (FMF) under measure transformations is critically important. It reveals the robustness of this class under key free probability operations like free convolution and various transformations and deepens our understanding of the connections between free cumulants, Voiculescu transforms, and convolution semigroups. Given that the FMF includes many fundamental distributions and possesses a rich structure, clarifying its stability enhances its utility in modeling, classification, and asymptotic analysis within random matrix theory and free probability.

The quadratic class of CSK families with

$${\mathcal{V}}_{\mu }\left(s\right)=1+\tau s+\varsigma s^2,\tau \in \mathbb{R},\varsigma \ge -1,m_0^{\mu }=0$$

(14)

were fully characterized in [5]. The relative distributions belong to the FMF:

$$\mu \left(dr\right)={\displaystyle \frac{\sqrt{4(1+\varsigma )-{(r-\tau )}^2}}{2\pi (\varsigma r^2+\tau r+1)}}{1}_{(\tau -2\sqrt{1+\varsigma },\tau +2\sqrt{1+\varsigma })}\left(r\right)dr+p_1{\delta }_{r_1}+p_2{\delta }_{r_2}.$$

(15)

We have the following:

(i)

If $\varsigma =0,{\tau }^2>1$, then $p_1=1-1/{\tau }^2,r_1=-1/\tau ,p_2=0$.

(ii)

If $\varsigma >0$ and ${\tau }^2>4\varsigma$, then $p_1=max\{0,1-{\displaystyle \frac{\left|\tau \right|-\sqrt{{\tau }^2-4\varsigma }}{2\varsigma \sqrt{{\tau }^2-4\varsigma }}}\},p_2=0$, and $r_1=\pm {\displaystyle \frac{\left|\tau \right|-\sqrt{{\tau }^2-4\varsigma }}{2\varsigma }}$, with the sign opposite to the sign of $\tau$.

(iii)

If $-1\le \varsigma <0$, then

$$r_{1,2}={\displaystyle \frac{-\tau \pm \sqrt{{\tau }^2-4\varsigma }}{2\varsigma }},p_{1,2}=1+{\displaystyle \frac{\sqrt{{\tau }^2-4\varsigma }\mp \tau }{2\varsigma \sqrt{{\tau }^2-4\varsigma }}}.$$

This result applies to several important classes of measures. Specifically, up to a dilation and a free convolution, the measure $\mu$ is

(i)

The semicircle ($\mathbb{SC}$) measure if $\tau =\varsigma =0$.

(ii)

The Marchenko–Pastur ($\mathbb{MP}$) measure if $\varsigma =0$ and $\tau \ne 0$.

(iii)

The free Pascal ($\mathbb{FP}$) measure if $\varsigma >0$ and ${\tau }^2>4\varsigma$.

(iv)

The free Gamma ($\mathbb{FG}$) measure if $\varsigma >0$ and ${\tau }^2=4\varsigma$.

(v)

The free analog of hyperbolic ($\mathbb{FH}$) measure if $\varsigma >0$ and ${\tau }^2<4\varsigma$.

(vi)

The free binomial ($\mathbb{FB}$) measure if $-1\le \varsigma <0$.

We then show that the FMF is closed under the ${\mathcal{U}}_{(a,t)}$-deformation. $D_{\alpha }\left(\rho \right)$ represents the dilation of a measure $\rho$ by $\alpha \ne 0$.

Theorem 2.

If $\rho \in \mathrm{FMF}$, then ∀$a\in \mathbb{R}$ and ∀$t>0$, ${\mathbf{T}}_{a{\mathit{r}}_2\left(\rho \right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\rho \right)\right)\right)\in \mathrm{FMF}$, where ${\mathbf{T}}_{\beta }\left(x\right)=x+\beta$ with $\beta \in \mathbb{R}$.

Proof. 

Assume that $\rho \in \mathrm{FMF}$. For $\mathcal{K}\left(\rho \right)$, the VF is

$${\mathcal{V}}_{\rho }\left(s\right)=1+{\tau }^{\prime }s+{\varsigma }^{\prime }{s}^2,{\tau }^{\prime }\in \mathbb{R},{\varsigma }^{\prime }\ge -1.$$

(16)

By combining (6), (16), and (10), one gets for s close to $m_0^{{\mathbf{T}}_{a{\mathbf{r}}_2\left(\rho \right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\rho \right)\right)\right)}=0$,

$${\mathcal{V}}_{{\mathbf{T}}_{a{\mathbf{r}}_2\left(\rho \right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\rho \right)\right)\right)}\left(s\right)=1+\left({\tau }^{\prime }\sqrt{t}+a\right)s+\left(t({\varsigma }^{\prime }+1)-1\right)s^2,$$

(17)

which is a VF of the kind specified in (14). Then, ${\mathbf{T}}_{a{\mathbf{r}}_2\left(\rho \right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\rho \right)\right)\right)\in \mathrm{FMF}$. □

Theorem 1 establishes that the FMF is closed under the ${\mathcal{U}}_{(a,t)}$-transformation. This non-trivial structural stability underscores the FMF’s fundamental role in non-commutative probability, justifying a deeper classification of these measures. We now illustrate the power of this theorem with concrete examples.

Corollary 1.

Consider $\mathit{sb}\left(dr\right)={\displaystyle \frac{1}{2}}{\delta }_{-1}+{\displaystyle \frac{1}{2}}{\delta }_1$. Then, ${\mathbf{T}}_{a{\mathit{r}}_2\left(\mathit{sb}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\mathit{sb}\right)\right)\right)$ is an $\mathbb{FB}$ measure of the form (15) with $\tau =a$ and $\varsigma =-1$.

Proof. 

For $\mathcal{K}\left(\mathbf{sb}\right)$, the VF is of the kind (16) for ${\tau }^{\prime }=0$ and ${\varsigma }^{\prime }=-1$. From (17) one has ${\mathcal{V}}_{{\mathbf{T}}_{a{\mathbf{r}}_2\left(\mathbf{sb}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\mathbf{sb}\right)\right)\right)}\left(s\right)=1+as-s^2$, for s close to 0. This compared to (14) ends the proof by (5). □

Corollary 2.

Consider the $\mathbb{SC}$ distribution

$${\displaystyle \mathit{sc}\left(dr\right)=\frac{1}{2\pi }\sqrt{4-r^2}{1}_{(-2,2)}\left(r\right)dr.}$$

(18)

Then, ${\mathbf{T}}_{a{\mathit{r}}_2\left(\mathit{sc}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\mathit{sc}\right)\right)\right)$ is a

(i) 

$\mathbb{SC}$ measure, if $t=1$ and $a=0$.

(ii) 

$\mathbb{MP}$ measure with $\tau =a$ and $\varsigma =0$ if, $t=1$ and $a\ne 0$.

(iii) 

$\mathbb{FP}$ measure with $\tau =a$ and $\varsigma =-1+t$, if $t>1$ and $a^2>4(-1+t)$.

(iv) 

$\mathbb{FG}$ measure with $\tau =a$ and $\varsigma =-1+t$, if $t>1$ and $a^2=4(-1+t)$.

(v) 

$\mathbb{FH}$ measure with $\tau =a$ and $\varsigma =-1+t$, if $t>1$ and $a^2<4(-1+t)$.

(vi) 

$\mathbb{FB}$ measure with $\tau =a$ and $\varsigma =-1+t$, if $0<t<1$.

Proof. 

For $\mathcal{K}\left(\mathbf{sc}\right)$, the VF is of the form (16) with ${\tau }^{\prime }={\varsigma }^{\prime }=0$. Based on (17), one has

$${\mathcal{V}}_{{\mathbf{T}}_{a{\mathbf{r}}_2\left(\mathbf{sc}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\mathbf{sc}\right)\right)\right)}\left(s\right)=1+as+\left(t-1\right)s^2,s\mathrm{close} \mathrm{to}0.$$

(19)

Comparing (19) and (14), one gets

(i)

If $t=1$ and $a=0$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left(\mathbf{sc}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\mathbf{sc}\right)\right)\right)=\mathbf{sc}$.

(ii)

If $t=1$ and $a\ne 0$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left(\mathbf{sc}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\mathbf{sc}\right)\right)\right)$ is an $\mathbb{MP}$ measure (15) with $\tau =a$ and $\varsigma =0$.

(iii)

If $t>1$ and $a^2>4(-1+t)$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left(\mathbf{sc}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\mathbf{sc}\right)\right)\right)$ is an $\mathbb{FP}$ measure (15) with $\tau =a$ and $\varsigma =-1+t$.

(iv)

If $t>1$ and $a^2=4(-1+t)$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left(\mathbf{sc}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\mathbf{sc}\right)\right)\right)$ is an $\mathbb{FG}$ measure (15) with $\tau =a$ and $\varsigma =-1+t$.

(v)

If $t>1$ and $a^2<4(-1+t)$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left(\mathbf{sc}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\mathbf{sc}\right)\right)\right)$ is an $\mathbb{FH}$ measure (15) with $\tau =a$ and $\varsigma =-1+t$.

(vi)

If $0<t<1$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left(\mathbf{sc}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\mathbf{sc}\right)\right)\right)$ is an $\mathbb{FB}$ measure (15) with $\tau =a$ and $\varsigma =t-1$.

□

Corollary 3.

For ${\tau }^{\prime }\ne 0$ and ${\varsigma }^{\prime }=0$, consider the $\mathbb{MP}$ measure

$${\mathit{mp}}_{{\tau }^{\prime }}\left(dr\right)={\displaystyle \frac{\sqrt{4-{(r-{\tau }^{\prime })}^2}}{2\pi ({\tau }^{\prime }r+1)}}{1}_{({\tau }^{\prime }-2,{\tau }^{\prime }+2)}\left(r\right)dr+{(1-1/{\tau }^{\prime 2})}^{+}{\delta }_{-1/{\tau }^{\prime }}.$$

(20)

Then, ${\mathbf{T}}_{a{\mathit{r}}_2\left({\mathit{mp}}_{{\tau }^{\prime }}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathit{mp}}_{{\tau }^{\prime }}\right)\right)\right)$ is a

(i) 

$\mathbb{SC}$ measure with $\tau ={\tau }^{\prime }+a=0$ and $\varsigma ={\varsigma }^{\prime }=0$ if $t=1$ and $a=-{\tau }^{\prime }$.

(ii) 

$\mathbb{MP}$ measure with $\tau ={\tau }^{\prime }+a$ and $\varsigma ={\varsigma }^{\prime }=0$ if $t=1$ and $a\ne -{\tau }^{\prime }$.

(iii) 

$\mathbb{FP}$ measure with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =-1+t$, if $t>1$ and ${({\tau }^{\prime }\sqrt{t}+a)}^2>4(-1+t)$.

(iv) 

$\mathbb{FG}$ measure with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =-1+t$, if $t>1$ and ${({\tau }^{\prime }\sqrt{t}+a)}^2=4(-1+t)$.

(v) 

$\mathbb{FH}$ measure with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =-1+t$, if $t>1$ and ${({\tau }^{\prime }\sqrt{t}+a)}^2<4(-1+t)$.

(vi) 

$\mathbb{FB}$ measure with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =-1+t$, if $0<t<1$.

Proof. 

For $\mathcal{K}\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)$, the VF is of the form (16) with ${\tau }^{\prime }\ne 0$ and ${\varsigma }^{\prime }=0$. Based on (17), one has

$${\mathcal{V}}_{{\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)\right)\right)}\left(s\right)=1+({\tau }^{\prime }\sqrt{t}+a)s+\left(t-1\right)s^2,s\mathrm{close}\mathrm{to}0.$$

(21)

Comparing (21) and (14) one gets the following:

(i)

If $t=1$ and $a=-{\tau }^{\prime }$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)\right)\right)$ is an $\mathbb{SC}$ measure (15) with $\tau ={\tau }^{\prime }+a=0$ and $\varsigma =0$.

(ii)

If $t=1$ and $a\ne -{\tau }^{\prime }$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)\right)\right)$ is an $\mathbb{MP}$ measure (15) with $\tau ={\tau }^{\prime }+a$ and $\varsigma =0$.

(iii)

If $t>1$ and ${({\tau }^{\prime }\sqrt{t}+a)}^2>4(-1+t)$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)\right)\right)$ is an $\mathbb{FP}$ measure (15) with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =-1+t$.

(iv)

If $t>1$ and ${({\tau }^{\prime }\sqrt{t}+a)}^2=4(-1+t)$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)\right)\right)$ is an $\mathbb{FG}$ measure (15) with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =-1+t$.

(v)

If $t>1$ and ${({\tau }^{\prime }\sqrt{t}+a)}^2<4(-1+t)$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)\right)\right)$ is an $\mathbb{FH}$ measure (15) with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =-1+t$.

(vi)

If $0<t<1$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{mp}}_{{\tau }^{\prime }}\right)\right)\right)$ is an $\mathbb{FB}$ measure (15) with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =-1+t$.

□

A notable feature of the ${\mathcal{U}}_{(a,t)}$-transformation is that, for appropriate parameters, it maps the $\mathbb{MP}$ measure onto the $\mathbb{SC}$ measure. This remarkable link between two central distributions of free probability that cannot be realized by the the t-transformation alone underscores the greater unifying power of the ${\mathcal{U}}_{(a,t)}$-transformation.

Corollary 4.

For ${\tau }^{\prime }\ne 0$ and ${\varsigma }^{\prime }={\tau }^{\prime 2}/4$, consider the $\mathbb{FG}$ measure

$${\mathit{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\left(dr\right)={\displaystyle \frac{\sqrt{4(1+{\varsigma }^{\prime })-{(r-{\tau }^{\prime })}^2}}{2\pi ({\varsigma }^{\prime }{r}^2+{\tau }^{\prime }r+1)}}{1}_{({\tau }^{\prime }-2\sqrt{1+{\varsigma }^{\prime }},{\tau }^{\prime }+2\sqrt{1+{\varsigma }^{\prime }})}\left(r\right)dr.$$

Then, ${\mathbf{T}}_{a{\mathit{r}}_2\left({\mathit{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathit{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)\right)$ is an

(i) 

$\mathbb{SC}$ measure with $\tau ={\tau }^{\prime }\sqrt{t}+a=0$ and $\varsigma =t({\tau }^{\prime 2}/4+1)-1=0$, if $t={\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}$ and $a=-{\tau }^{\prime }\sqrt{{\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}}$.

(ii) 

$\mathbb{MP}$ measure with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =t({\tau }^{\prime 2}/4+1)-1=0$, if $t={\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}$ and $a\ne -{\tau }^{\prime }\sqrt{{\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}}$.

(iii) 

$\mathbb{FP}$ measure with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =t({\tau }^{\prime 2}/4+1)-1$, if $t>{\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}$ and ${({\tau }^{\prime }\sqrt{t}+a)}^2>t{\tau }^{\prime 2}+4(-1+t)$.

(iv) 

$\mathbb{FG}$ measure with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =t({\tau }^{\prime 2}/4+1)-1$, if $t>{\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}$ and ${({\tau }^{\prime }\sqrt{t}+a)}^2=t{\tau }^{\prime 2}+4(-1+t)$.

(v) 

$\mathbb{FH}$ measure with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =t({\tau }^{\prime 2}/4+1)-1$, if $t>{\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}$ and ${({\tau }^{\prime }\sqrt{t}+a)}^2<t{\tau }^{\prime 2}+4(-1+t)$.

(vi) 

$\mathbb{FB}$ measure with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =t({\tau }^{\prime 2}/4+1)-1$, if $t<{\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}$.

Proof. 

For $\mathcal{K}\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)$, the VF is of the form (16) with ${\tau }^{\prime }\ne 0$ and ${\varsigma }^{\prime }={\tau }^{\prime 2}/4$. Based on (17), one has

$${\mathcal{V}}_{{\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)\right)}\left(s\right)=1+({\tau }^{\prime }\sqrt{t}+a)s+\left(t({\tau }^{\prime 2}/4+1)-1\right)s^2,s\mathrm{close}\mathrm{to}0.$$

(22)

Comparing (22) and (14) one gets the following:

(i)

If $t={\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}$ and $a=-{\tau }^{\prime }\sqrt{{\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}}$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)\right)$ is an $\mathbb{SC}$ measure (15) with $\tau ={\tau }^{\prime }\sqrt{t}+a=0$ and $\varsigma =t({\tau }^{\prime 2}/4+1)-1=0$.

(ii)

If $t={\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}$ and $a\ne -{\tau }^{\prime }\sqrt{{\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}}$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)\right)$ is an $\mathbb{MP}$ measure (15) with $\tau ={\tau }^{\prime }\sqrt{{\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}}+a$ and $\varsigma =t({\tau }^{\prime 2}/4+1)-1=0$.

(iii)

If $t>{\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}$ and ${({\tau }^{\prime }\sqrt{t}+a)}^2>t{\tau }^{\prime 2}+4(-1+t)$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)\right)$ is an $\mathbb{FP}$ measure (15) with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =t({\tau }^{\prime 2}/4+1)-1$.

(iv)

If $t>{\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}$ and ${({\tau }^{\prime }\sqrt{t}+a)}^2=t{\tau }^{\prime 2}+4(-1+t)$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)\right)$ is an $\mathbb{FG}$ measure (15) with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =t({\tau }^{\prime 2}/4+1)-1$. The case where $t=1$ and $a=0$ is included in this possibility.

(v)

If $t>{\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}$ and ${({\tau }^{\prime }\sqrt{t}+a)}^2<t{\tau }^{\prime 2}+4(-1+t)$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)\right)$ is an $\mathbb{FH}$ measure (15) with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =t({\tau }^{\prime 2}/4+1)-1$.

(vi)

If $t<{\displaystyle \frac{1}{1+{\tau }^{\prime 2}/4}}$, then ${\mathbf{T}}_{a{\mathbf{r}}_2\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)}\left({\mathcal{U}}_{(a,1/t)}\left(D_{\sqrt{t}}\left({\mathbf{fg}}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)\right)$ is an $\mathbb{FB}$ measure (15) with $\tau ={\tau }^{\prime }\sqrt{t}+a$ and $\varsigma =t({\tau }^{\prime 2}/4+1)-1$.

□

Note that one can recover the $\mathbb{SC}$ measure (or the $\mathbb{MP}$ measure) from a ${\mathcal{U}}_{(a,t)}$-transformation of the $\mathbb{FG}$ measure, something not achievable by the t-transform alone. This result highlights the strength of the newly introduced ${\mathcal{U}}_{(a,t)}$-transformation. Starting from the $\mathbb{FG}$ measure, one can recover the $\mathbb{SC}$ measure (or the $\mathbb{MP}$ measure) as a particular case by tuning the parameters of ${\mathcal{U}}_{(a,t)}$. The unified operator ${\mathcal{U}}_{(a,t)}$ thus provides a broader framework in which central distributions of free probability—$\mathbb{SC}$, $\mathbb{MP}$, and $\mathbb{FG}$—can be linked through a single transformation.

5. Limit Theorems Involving ${\mathcal{U}}_{(\mathit{a},\mathit{t})}$-Transformation

We present new limiting theorems involving ${\mathcal{U}}_{(a,t)}$-transformation and including the free additive convolution ⊞ (see [10,11]) and the Belinschi–Nica transformation of measures [12]

$$\begin{array}{ccc}\hfill {\mathbb{B}}_l:\mathbb{P}& \to & \mathbb{P};\rho \mapsto {\mathbb{B}}_l\left(\rho \right)={\left({\rho }^{⊞(1+l)}\right)}^{\uplus {\displaystyle \frac{1}{1+l}}},l\ge 0.\hfill \end{array}$$

For $l=1$, the transformation ${\mathbb{B}}_1$ represents the Bercovici and Pata bijection.

Let $\rho \in {\mathbb{P}}_{ba}$ with finite mean. For $\lambda >0$, so that ${\rho }^{⊞\lambda }$ is defined, and for s close sufficiently to $m_0^{{\rho }^{⊞\lambda }}=\lambda m_0^{\rho }$ [7],

$${\mathcal{V}}_{{\rho }^{⊞\lambda }}\left(s\right)=s{\mathcal{V}}_{\rho }(s/\lambda ).$$

(23)

Furthermore, for $l\ge 0$ and for s close to $m_0^{\rho }$ [9],

$${\mathcal{V}}_{{\mathbb{B}}_l\left(\rho \right)}\left(s\right)={\mathcal{V}}_{\rho }\left(s\right)+ls(s-m_0^{\rho })\mathrm{and}{\mathcal{V}}_{{\mathbb{B}}_1^{-1}\left(\rho \right)}\left(s\right)={\mathcal{V}}_{\rho }\left(s\right)-s(s-m_0^{\rho }).$$

(24)

We next state and demonstrate the section’s results.

Theorem 3.

Let $\rho \in {\mathcal{P}}_c$. Then,

(i) 

For $a\in \mathbb{R}$ and $t>0$, so that measure ${\mathcal{U}}_{(a,t)}\left({\rho }^{⊞1/t}\right)$ is well defined, one has

$${\mathcal{U}}_{(a,t)}\left({\rho }^{⊞1/t}\right)\stackrel{t\to +\infty }{\to }{V}_a\left({\mathbb{B}}_1^{-1}\left(\rho \right)\right),indistribution.$$

(25)

(ii) 

For $a\in \mathbb{R}$ and $t>0$, so that measure ${\left({\mathcal{U}}_{(a,1/t)}\left(\rho \right)\right)}^{⊞t}$ is well defined, one has

$${\left({\mathcal{U}}_{(a,1/t)}\left(\rho \right)\right)}^{⊞t}\stackrel{t\to +\infty }{\to }{\mathbb{B}}_1\left(V_a\left(\rho \right)\right),indistribution.$$

(26)

Proof. 

(i) Using (23) and (10), for s close sufficiently to $m_0^{{\mathcal{U}}_{(a,t)}\left({\rho }^{⊞1/t}\right)}=t{m}_0^{{\rho }^{⊞1/t}}-at{\mathbf{r}}_2\left({\rho }^{⊞1/t}\right)=$$m_0^{\rho }-a{\mathbf{r}}_2\left(\rho \right)$, one has

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{{\mathcal{U}}_{(a,t)}\left({\rho }^{⊞1/t}\right)}\left(s\right)& =& t{\mathcal{V}}_{{\rho }^{⊞1/t}}\left({\displaystyle \frac{s}{t}}+a{\mathbf{r}}_2\left({\rho }^{⊞1/t}\right)\right)+\left({\displaystyle \frac{s}{t}}+a{\mathbf{r}}_2\left({\rho }^{⊞1/t}\right)-s\right)\left(s+at{\mathbf{r}}_2\left({\rho }^{⊞1/t}\right)-t{m}_0^{{\rho }^{⊞1/t}}\right)\hfill \\ & =& t{\mathcal{V}}_{{\rho }^{⊞1/t}}\left({\displaystyle \frac{s+a{\mathbf{r}}_2\left(\rho \right)}{t}}\right)+\left({\displaystyle \frac{s+a{\mathbf{r}}_2\left(\rho \right)}{t}}-s\right)\left(s+a{\mathbf{r}}_2\left(\rho \right)-m_0^{\rho }\right)\hfill \\ & =& {\mathcal{V}}_{\rho }(s+a{\mathbf{r}}_2\left(\rho \right))+\left({\displaystyle \frac{s+a{\mathbf{r}}_2\left(\rho \right)}{t}}-s\right)\left(s+a{\mathbf{r}}_2\left(\rho \right)-m_0^{\rho }\right)\hfill \\ & \stackrel{t\to +\infty }{\to }& {\mathcal{V}}_{\rho }(s+a{\mathbf{r}}_2\left(\rho \right))-s\left(s+a{\mathbf{r}}_2\left(\rho \right)-m_0^{\rho }\right).\hfill \end{array}$$

(27)

On the other hand, using (24) and (13), for s close enough to $m_0^{V_a\left({\mathbb{B}}_1^{-1}\left(\rho \right)\right)}=m_0^{{\mathbb{B}}_1^{-1}\left(\rho \right)}-a{\mathbf{r}}_2\left({\mathbb{B}}_1^{-1}\left(\rho \right)\right)=m_0^{\rho }-a{\mathbf{r}}_2\left(\rho \right)$, one has

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{V_a\left({\mathbb{B}}_1^{-1}\left(\rho \right)\right)}\left(s\right)& =& {\mathcal{V}}_{{\mathbb{B}}_1^{-1}\left(\rho \right)}(s+a{\mathbf{r}}_2\left({\mathbb{B}}_1^{-1}\left(\rho \right)\right))+a{\mathbf{r}}_2\left({\mathbb{B}}_1^{-1}\left(\rho \right)\right)\left(s+a{\mathbf{r}}_2\left({\mathbb{B}}_1^{-1}\left(\rho \right)\right)-m_0^{{\mathbb{B}}_1^{-1}\left(\rho \right)}\right)\hfill \\ & =& {\mathcal{V}}_{{\mathbb{B}}_1^{-1}\left(\rho \right)}(s+a{\mathbf{r}}_2\left(\rho \right))+a{\mathbf{r}}_2\left(\rho \right)\left(s+a{\mathbf{r}}_2\left(\rho \right)-m_0^{\rho }\right)\hfill \\ & =& {\mathcal{V}}_{\rho }(s+a{\mathbf{r}}_2\left(\rho \right))-(s+a{\mathbf{r}}_2\left(\rho \right))\left(s+a{\mathbf{r}}_2\left(\rho \right)-m_0^{\rho }\right)+a{\mathbf{r}}_2\left(\rho \right)\left(s+a{\mathbf{r}}_2\left(\rho \right)-m_0^{\rho }\right)\hfill \\ & =& {\mathcal{V}}_{\rho }(s+a{\mathbf{r}}_2\left(\rho \right))-s\left(s+a{\mathbf{r}}_2\left(\rho \right)-m_0^{\rho }\right).\hfill \end{array}$$

(28)

From (27) and (28), one sees that

$${\mathcal{V}}_{{\mathcal{U}}_{(a,t)}\left({\rho }^{⊞1/t}\right)}\left(s\right)\stackrel{t\to +\infty }{\to }{\mathcal{V}}_{V_a\left({\mathbb{B}}_1^{-1}\left(\rho \right)\right)}\left(s\right),s\mathrm{close}\mathrm{enough}\mathrm{to}{m}_0^{\rho }-a{\mathbf{r}}_2\left(\rho \right).$$

This result, combined with Proposition 4.2 of [5], achieves the demonstration of (25).

(ii) Using (23) and (10), for s near $m_0^{{\left({\mathcal{U}}_{(a,1/t)}\left(\rho \right)\right)}^{⊞t}}=t{m}_0^{{\mathcal{U}}_{(a,1/t)}\left(\rho \right)}=m_0^{\rho }-a{\mathbf{r}}_2\left(\rho \right)$, one has

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{{\left({\mathcal{U}}_{(a,1/t)}\left(\rho \right)\right)}^{⊞t}}\left(s\right)& =& t{\mathcal{V}}_{{\mathcal{U}}_{(a,1/t)}\left(\rho \right)}\left({\displaystyle \frac{s}{t}}\right)\hfill \\ & =& t\left[{\displaystyle \frac{1}{t}}{\mathcal{V}}_{\rho }(s+a{\mathbf{r}}_2\left(\rho \right))+\left(s+a{\mathbf{r}}_2\left(\rho \right)-{\displaystyle \frac{s}{t}}\right)\left({\displaystyle \frac{s+a{\mathbf{r}}_2\left(\rho \right)-m_0^{\rho }}{t}}\right)\right]\hfill \\ & =& {\mathcal{V}}_{\rho }(s+a{\mathbf{r}}_2\left(\rho \right))+\left(s+a{\mathbf{r}}_2\left(\rho \right)-{\displaystyle \frac{s}{t}}\right)\left(s+a{\mathbf{r}}_2\left(\rho \right)-m_0^{\rho }\right)\hfill \\ & \stackrel{t\to +\infty }{\to }& {\mathcal{V}}_{\rho }(s+a{\mathbf{r}}_2\left(\rho \right))+(s+a{\mathbf{r}}_2\left(\rho \right))\left(s+a{\mathbf{r}}_2\left(\rho \right)-m_0^{\rho }\right).\hfill \end{array}$$

(29)

On the other hand, using (24) and (13), for s close enough to $m_0^{{\mathbb{B}}_1\left(V_a\left(\rho \right)\right)}=m_0^{V_a\left(\rho \right)}=m_0^{\rho }-a{\mathbf{r}}_2\left(\rho \right)$, one has

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{{\mathbb{B}}_1\left(V_a\left(\rho \right)\right)}\left(s\right)& =& {\mathcal{V}}_{V_a\left(\rho \right)}\left(s\right)+s\left(s-m_0^{V_a\left(\rho \right)}\right)\hfill \\ & =& {\mathcal{V}}_{\rho }\left(s\right)+a{\mathbf{r}}_2\left(\rho \right)\left(s+a{\mathbf{r}}_2\left(\rho \right)-m_0^{\rho }\right)+s\left(s+a{\mathbf{r}}_2\left(\rho \right)-m_0^{\rho }\right)\hfill \\ & =& {\mathcal{V}}_{\rho }(s+a{\mathbf{r}}_2\left(\rho \right))+(s+a{\mathbf{r}}_2\left(\rho \right))\left(s+a{\mathbf{r}}_2\left(\rho \right)-m_0^{\rho }\right).\hfill \end{array}$$

(30)

From (29) and (30), one see that

$${\mathcal{V}}_{{\left({\mathcal{U}}_{(a,1/t)}\left(\rho \right)\right)}^{⊞t}}\left(s\right)\stackrel{t\to +\infty }{\to }{\mathcal{V}}_{{\mathbb{B}}_1\left(V_a\left(\rho \right)\right)}\left(s\right),s\mathrm{close}\mathrm{enough}\mathrm{to}{m}_0^{\rho }-a{\mathbf{r}}_2\left(\rho \right).$$

This result, combined with [5] (Proposition 4.2), achieves the demonstration of (26). □

Next, we highlight the importance of Theorem 3 by applying it to some specific measures.

Example 1.

Consider the $\mathbb{SC}$ measure (18); then, for $a\in \mathbb{R}$, we have

$${\mathcal{U}}_{(a,t)}\left({\mathit{sc}}^{⊞1/t}\right)\stackrel{t\to +\infty }{\to }{\mathit{T}}_{-a}\left({\mathit{fb}}_{(a,-1)}\right),indistribution,$$

where ${\mathit{fb}}_{(\tau ,\varsigma )}$ is the $\mathbb{FB}$ measure (15) with parameters $\tau \in \mathbb{R}$ and $-1\le \varsigma <0$. In fact, for s near $m_0^{{\mathcal{U}}_{(a,t)}\left({\mathit{sc}}^{⊞1/t}\right)}=-a$, one has

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{{\mathcal{U}}_{(a,t)}\left({\mathit{sc}}^{⊞1/t}\right)}\left(s\right)\stackrel{t\to +\infty }{\to }{\mathcal{V}}_{V_a\left({\mathbb{B}}_1^{-1}\left(\mathit{sc}\right)\right)}\left(s\right)& =& {\mathcal{V}}_{{\mathbb{B}}_1^{-1}\left(\mathit{sc}\right)}(s+a)+a(s+a)\hfill \\ & =& {\mathcal{V}}_{\mathit{sc}}(s+a)-{(s+a)}^2+a(s+a)\hfill \\ & =& 1+a(s+a)-{(s+a)}^2={\mathcal{V}}_{{\mathit{T}}_{-a}\left({\mathit{fb}}_{(a,-1)}\right)}\left(s\right).\hfill \end{array}$$

In particular, for $a=0$, one has

$${\mathcal{U}}_{(0,t)}\left({\mathit{sc}}^{⊞1/t}\right)=U_t\left({\mathit{sc}}^{⊞1/t}\right)={\left({\mathit{sc}}^{⊞1/t}\right)}^{\uplus t}\stackrel{t\to +\infty }{\to }\mathit{sb},indistribution.$$

Example 2.

Consider the symmetric Bernoulli measure $\mathit{sb}$; then, for $a\in \mathbb{R}$, we have

$${\left({\mathcal{U}}_{(a,1/t)}\left(\mathit{sb}\right)\right)}^{⊞t}\stackrel{t\to +\infty }{\to }{\mathit{T}}_{-a}\left(\mathit{sc}\right),indistribution,$$

Indeed, for s near $m_0^{{\left({\mathcal{U}}_{(a,1/t)}\left(\mathit{sb}\right)\right)}^{⊞t}}=-a$, one has

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{{\left({\mathcal{U}}_{(a,1/t)}\left(\mathit{sb}\right)\right)}^{⊞t}}\left(s\right)\stackrel{t\to +\infty }{\to }{\mathcal{V}}_{{\mathbb{B}}_1\left(V_a\left(\mathit{sb}\right)\right)}\left(s\right)& =& {\mathcal{V}}_{V_a\left(\mathit{sb}\right)}\left(s\right)+s(s+a)\hfill \\ & =& {\mathcal{V}}_{\mathit{sb}}(s+a)+a(s+a)+s(s+a)\hfill \\ & =& 1-{(s+a)}^2+{(s+a)}^2=1={\mathcal{V}}_{{\mathit{T}}_{-a}\left(\mathit{sc}\right)}\left(s\right).\hfill \end{array}$$

In particular, for $a=0$, one has

$${\left({\mathcal{U}}_{(0,1/t)}\left(\mathit{sb}\right)\right)}^{⊞t}={\left(U_{1/t}\left(\mathit{sb}\right)\right)}^{⊞t}={\left({\mathit{sb}}^{\uplus 1/t}\right)}^{⊞t}\stackrel{t\to +\infty }{\to }\mathit{sc},indistribution.$$

Example 3.

Consider the $\mathbb{MP}$ measure (20) with ${\tau }^{\prime }\ne 0$. Then, for $a\in \mathbb{R}$, we have

$${\mathcal{U}}_{(a,t)}\left({\mathit{mp}}_{{\tau }^{\prime }}^{⊞1/t}\right)\stackrel{t\to +\infty }{\to }{\mathit{T}}_{-a}\left({\mathit{fb}}_{({\tau }^{\prime }+a,-1)}\right),indistribution.$$

Indeed, for s near $m_0^{{\mathcal{U}}_{(a,t)}\left({\mathit{mp}}_{{\tau }^{\prime }}^{⊞1/t}\right)}=-a$, one has

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{{\mathcal{U}}_{(a,t)}\left({\mathit{mp}}_{{\tau }^{\prime }}^{⊞1/t}\right)}\left(s\right)\stackrel{t\to +\infty }{\to }{\mathcal{V}}_{V_a\left({\mathbb{B}}_1^{-1}\left({\mathit{mp}}_{{\tau }^{\prime }}\right)\right)}\left(s\right)& =& {\mathcal{V}}_{{\mathbb{B}}_1^{-1}\left({\mathit{mp}}_{{\tau }^{\prime }}\right)}(s+a)+a(s+a)\hfill \\ & =& {\mathcal{V}}_{{\mathit{mp}}_{{\tau }^{\prime }}}(s+a)-{(s+a)}^2+a(s+a)\hfill \\ & =& 1+{\tau }^{\prime }(s+a)+a(s+a)-{(s+a)}^2\hfill \\ & =& 1+({\tau }^{\prime }+a)(s+a)-{(s+a)}^2={\mathcal{V}}_{{\mathit{T}}_{-a}\left({\mathit{fb}}_{({\tau }^{\prime }+a,-1)}\right)}\left(s\right).\hfill \end{array}$$

In particular, for $a=0$, one has

$${\mathcal{U}}_{(0,t)}\left({\mathit{mp}}_{{\tau }^{\prime }}^{⊞1/t}\right)=U_t\left({\mathit{mp}}_{{\tau }^{\prime }}^{⊞1/t}\right)={\left({\mathit{mp}}_{{\tau }^{\prime }}^{⊞1/t}\right)}^{\uplus t}\stackrel{t\to +\infty }{\to }{\mathit{fb}}_{({\tau }^{\prime },-1)},indistribution,$$

and for $a=-{\tau }^{\prime }$, one has

$${\mathcal{U}}_{(-{\tau }^{\prime },t)}\left({\mathit{mp}}_{{\tau }^{\prime }}^{⊞1/t}\right)\stackrel{t\to +\infty }{\to }{\mathit{T}}_{{\tau }^{\prime }}\left(\mathit{sb}\right),indistribution.$$

Our limiting theorems use the ${\mathcal{U}}_{(a,t)}$-transformation to bridge cornerstone measures of free probability, specifically the $\mathbb{SC}$, $\mathbb{MP}$, and $\mathbb{FB}$ laws. This connection reveals a unified hierarchical structure that was not apparent from studying the $V_a$-transformation separately.

6. Conclusions

This work underscores the profound utility of CSK families and their VFs as a powerful framework for analyzing measure transformations. We have introduced and systematically analyzed the ${\mathcal{U}}_{(a,t)}$-transformation, a novel operation on probability measures that provides a unifying framework for two pivotal transformations in free probability: the t-transformation and the $V_a$-transformation. By defining this transformation through a modified functional equation for the Cauchy transform, we have established a robust analytic foundation that seamlessly generalizes these previously distinct concepts. Our investigation from the perspective of CSK families and their associated VFs has proven to be exceptionally fruitful. We successfully derived a general expression for the VF of a CSK family generated by a measure transformed by ${\mathcal{U}}_{(a,t)}$, a result that serves as a powerful computational engine for exploring the transformation’s effects. This formula was instrumental in proving a central finding of this paper: the FMF is invariant under the ${\mathcal{U}}_{(a,t)}$-transformation. This invariance solidifies the free Meixner class as a natural and fundamental object within this expanded operational calculus, much like its classical counterparts are in similar contexts. Furthermore, the limiting theorems we established demonstrate the significant power of the ${\mathcal{U}}_{(a,t)}$-transformation to generate new insights and connections within the menagerie of free probability. These results illuminate a clear and structured pathway between cornerstone measures such as the $\mathbb{SC}$, $\mathbb{MP}$, and $\mathbb{FB}$ laws, revealing a deeper hierarchical structure that was previously obscured by treating their transformations separately.

Looking forward, this work opens up several compelling avenues for research. In [1,2], a new type of convolution termed t-transformation of free convolution, denoted by $\begin{array}{|c|}\hline t\\ \hline\end{array}$, is presented, based on the t-deformation of measures. For $\sigma$ and $\lambda \in \mathbb{P}$, the t-deformed free convolution is defined as

$$\sigma \begin{array}{|c|}\hline t\\ \hline\end{array}\lambda =U_{1/t}(U_t\left(\sigma \right)⊞U_t\left(\lambda \right))$$

Furthermore, based on $V_a$-deformation of measures, another sort of convolution termed $V_a$-deformed free convolution, indicated by $\begin{array}{|c|}\hline a\\ \hline\end{array}$, is introduced in [3]. If $\rho$ and $\varrho \in \mathbb{P}$ have finite second moments, their $V_a$-transformed free convolution is introduced as

$$\rho \begin{array}{|c|}\hline a\\ \hline\end{array}\varrho =V_{-a}(V_a\left(\rho \right)⊞V_a\left(\varrho \right)).$$

Note that for $a\in \mathbb{R}$ and $t>0$, the inverse of ${\mathcal{U}}_{(a,t)}$ is ${\mathcal{U}}_{(a,t)}^{-1}={\mathcal{U}}_{(-a,1/t)}$. In fact, consider $\rho \in \mathbb{P}$ having finite variance. Let $a_1$, $a_2$ be real numbers and $t_1$, $t_2$ be strictly positive real numbers such that ${\mathcal{U}}_{(a_1,t_1)}\left({\mathcal{U}}_{(a_2,t_2)}\left(\rho \right)\right)=\rho$. By the use of (1), we get

$${\displaystyle \frac{1}{{\mathcal{G}}_{\rho }\left(z\right)}}={\displaystyle \frac{1}{{\mathcal{G}}_{{\mathcal{U}}_{(a_1,t_1)}\left({\mathcal{U}}_{(a_2,t_2)}\left(\rho \right)\right)}\left(z\right)}}={\displaystyle \frac{t_1{t}_2}{{\mathcal{G}}_{\rho }\left(z\right)}}+(1-t_1{t}_2)z+t_1{t}_2{\mathbf{r}}_2\left(\rho \right)[a_1+a_2].$$

This implies that $a_2=-a_1$ and $t_2=1/t_1$.

By considering $a\in \mathbb{R}$ and $t>0$ and based on the ${\mathcal{U}}_{(a,t)}$-transformation of measures, one may introduce a new kind of convolution which we call ${\mathcal{U}}_{(a,t)}$-deformed free convolution and we denote by ${⊞}_{{\mathcal{U}}_{(a,t)}}$: if $\rho$ and $\varrho \in \mathbb{P}$ have finite variances, we define their ${\mathcal{U}}_{(a,t)}$-deformed free convolution as

$$\rho {⊞}_{{\mathcal{U}}_{(a,t)}}\varrho ={\mathcal{U}}_{(-a,1/t)}({\mathcal{U}}_{(a,t)}\left(\rho \right)⊞{\mathcal{U}}_{(a,t)}\left(\varrho \right)),a\in \mathbb{R}\mathrm{and}t>0.$$

(31)

It is worth mentioning that for $a=0$, we have ${⊞}_{{\mathcal{U}}_{(a,t)}}=\begin{array}{|c|}\hline t\\ \hline\end{array}$, and for $t=1$, we obtain ${⊞}_{{\mathcal{U}}_{(a,t)}}=\begin{array}{|c|}\hline a\\ \hline\end{array}$. The introduction of the ${⊞}_{{\mathcal{U}}_{(a,t)}}$-convolution opens up a new frontier for research in free probability. We anticipate that its further study will not only provide a deeper unification of the theory but also facilitate the discovery of novel limiting distributions and strengthen the connections to random matrix theory and operator algebras.