A Study of the (a, b)-Deformed Free Convolution

Abstract

This study is devoted to the detailed examination of the concept of $(a,b)$-deformation, defined for parameters $a\in \mathbb{R}$ and $b>0$. The analysis is conducted within the framework of Cauchy–Stieltjes kernel (CSK) families of probability measures, with particular attention given to the role of their variance functions (VFs). Using the VF as the main analytical tool, it is shown that the $(a,b)$-deformation of any measure belonging to the free Meixner family (FMF) remains within the same family. Moreover, the VF framework provides a powerful and flexible means for establishing new limit theorems associated with $(a,b)$-deformed free convolution. In particular, several novel limiting behaviors are derived, which naturally encompass both free and Boolean additive convolutions as special cases.

1. Introduction

A central objective of probability theory is to describe and analyze how the distributions of random variables evolve under various operations and transformations. Understanding these changes is fundamental for characterizing the stability, structure, and asymptotic behavior of probability measures. In this regard, two fundamental notions, measure transformations and convolution operations, play a key role in both classical and modern probability theory. These concepts not only provide deep theoretical insight but also serve as essential tools in applied domains such as stochastic modeling, signal processing, and statistical inference [1,2,3]. Their study has led to the development of numerous analytical frameworks and deformation schemes [4,5,6], each contributing to a deeper understanding of how algebraic operations interact with probabilistic structures, particularly in non-commutative and generalized settings.

Recent progress in free and Boolean probability has further emphasized the importance of analytic transformations, such as the t-transformation [7,8] and the $V_a$-transformation [9], in describing deformation and invariance phenomena of probability measures. Within this context, Cauchy–Stieltjes kernel (CSK) families provide a unifying analytical framework that connects transform-based methods with statistical structures such as mean and variance functions (VFs). However, most existing works focus on single–parameter deformations, which limits our understanding of how multi-parameter transformations influence the analytic, geometric, and statistical properties of CSK families.

This paper aims to fill this gap by investigating the $(a,b)$-transformation, $a\in \mathbb{R}$ and $b>0$, a two-parameter deformation originally proposed in [10]. This transformation offers a flexible mechanism for deforming probability measures while preserving key analytical and probabilistic structures. Its formulation allows the exploration of symmetries, invariance properties, and deformation behaviors arising from parameter variations, thus enriching the theory of measure transformations within both classical and free probabilistic frameworks. The approach developed in this work extends previous one–dimensional analysis to a richer, two-dimensional setting. Through this analysis, we provide a unified analytic and statistical interpretation of the $(a,b)$-transformation, bridging the gap between algebraic deformation theory and the probabilistic geometry of non–commutative distributions, and revealing new layers of symmetry within the CSK families framework.

Let $\mathcal{P}$ denote the set of all probability measures on the real line. Consider a measure $\eta \in \mathcal{P}$ with a finite first moment. For parameters $a\in \mathbb{R}$ and $b>0$, let us write $\mathbf{t}=(a,b)$, and define the $\mathbf{t}$-deformation of $\eta$ by

$${\displaystyle \frac{1}{{\mathcal{G}}_{{\tilde{U}}^{\mathbf{t}}\left(\eta \right)}\left(\varpi \right)}}={\displaystyle \frac{b}{{\mathcal{G}}_{\eta }\left(\varpi \right)}}+(1-b)\varpi +(b-a)m_0^{\eta },$$

where $m_0^{\eta }$ denotes the first moment of $\eta$. Here,

$${\mathcal{G}}_{\eta }\left(\varpi \right)=\int {\displaystyle \frac{\eta \left(ds\right)}{\varpi -s}},\varpi \in \mathbb{C}\setminus \mathrm{supp}\left(\eta \right)$$

is the Cauchy–Stieltjes transform (CST) of $\eta$, and $\mathrm{supp}\left(\eta \right)$ denotes its support.

It is worth noting that when the parameters satisfy $t=a=b>0$, the transformation ${\tilde{U}}^{\mathbf{t}}(·)$ coincides with the original t-deformation introduced in [7,8].

The $\mathbf{t}$-deformation of measures admits another elegant description in terms of continued fractions. If the measure $\eta$ has finite moments of all orders, then its CST admits the continued fraction expansion

$${\displaystyle {\mathcal{G}}_{\eta }\left(\varpi \right)=\frac{1}{{\displaystyle \varpi -{\kappa }_1-\frac{{\iota }_1}{{\displaystyle \varpi -{\kappa }_2-\frac{{\iota }_2}{{\displaystyle \varpi -{\kappa }_3-\frac{{\iota }_3}{\varpi -{\kappa }_4-\dots }}}}}}},}$$

where ${\kappa }_j$ and ${\iota }_j$ denote the recursion coefficients. In this representation, the $\mathbf{t}$-deformation simply acts by multiplying the coefficients ${\kappa }_1$ and ${\iota }_1$ by a and b, respectively. More explicitly,

$${\displaystyle {\mathcal{G}}_{{\tilde{U}}^{\mathbf{t}}\left(\eta \right)}\left(\varpi \right)=\frac{1}{{\displaystyle \varpi -a{\kappa }_1-\frac{b{\iota }_1}{{\displaystyle \varpi -{\kappa }_2-\frac{{\iota }_2}{{\displaystyle \varpi -{\kappa }_3-\frac{{\iota }_3}{\varpi -{\kappa }_4-\dots }}}}}}}.}$$

This characterization makes the deformation particularly transparent and connects it to the combinatorial structure of orthogonal polynomials.

The notion of $\mathbf{t}$-deformation naturally extends to the level of free convolution. As established in [10] (Proposition 1.4), the operator ${\tilde{U}}^{\mathbf{t}}$ is invertible whenever $b>0$ and $a\ne 0$. Its inverse is again of the same form, namely ${\tilde{U}}^{{\mathbf{t}}^{-1}}$, where ${\mathbf{t}}^{-1}=(1/a,1/b)$. This property allows one to define the $\mathbf{t}$-deformed free convolution ${⊞}_{\left(\mathbf{t}\right)}$ by [10]

$$\eta {⊞}_{\left(\mathbf{t}\right)}\lambda ={\tilde{U}}^{{\mathbf{t}}^{-1}}({\tilde{U}}^{\mathbf{t}}\left(\eta \right)⊞{\tilde{U}}^{\mathbf{t}}\left(\lambda \right)),$$

for any probability measures $\eta ,\lambda \in \mathcal{P}$ having finite first moments.

The central limiting behavior for the ${⊞}_{\left(\mathbf{t}\right)}$-convolution parallels that of the t-deformation. In particular, the central limit measure is the same in both cases. However, for the Poisson limit, the situation is different: the limiting measure in the ${⊞}_{\left(\mathbf{t}\right)}$-framework depends explicitly on the two parameters $(a,b)$. The corresponding Poisson limit theorem can be computed explicitly, and the limiting law is described in terms of orthogonal polynomials. Beyond these results, further developments have been obtained in [11], where the ${⊞}_{\left(\mathbf{t}\right)}$-convolution is analyzed within the framework of CSK families and their VFs. In that context, the VF is explicitly expressed in terms of ${⊞}_{\left(\mathbf{t}\right)}$-convolution powers. Moreover, specific estimates are derived for the $\mathbf{t}$-transformed free Gaussian CSK family as well as for the $\mathbf{t}$-transformed free Poisson CSK family. Furthermore, by incorporating the free multiplicative convolution, a new limiting theorem is established for the ${⊞}_{\left(\mathbf{t}\right)}$-convolution. This result highlights the flexibility of the $\mathbf{t}$-deformation in generating novel asymptotic behaviors and underlines its relevance in the broader setting of free probability theory.

This work represents a continuation of the study of the $(a,b)$-transformation within the framework of CSK families, following the initial developments presented in [11]. The choice of this topic is motivated by the central importance of measure deformations and convolutions in several fields, including statistical inference, probability theory, and finance. These operations provide the mathematical tools necessary to model structural changes, analyze random systems, and capture the evolution of distributions under different types of transformations. However, in probability theory, two complementary directions of study are particularly relevant. On the one hand, limit theorems describe the long-term or asymptotic behavior of sequences of random variables or measures [12,13,14]. On the other hand, stability analysis ensures that certain families of measures preserve their defining features when subjected to deformations or transformations. In this regard, a natural question arises: does the free Meixner family (FMF) remain stable under $(a,b)$-transformations? Answering this question is essential because the FMF plays a central role in free probability, serving as an analogue of classical Meixner families. The novelty of this research lies in two main aspects. First, it addresses the problem of stability of the FMF under $(a,b)$-transformations, which has not been fully explored in the literature. Second, it introduces a methodological innovation by employing the VF of CSK families as the principal analytical tool. The VF not only provides an elegant framework for analyzing stability but also enables the derivation of new limit theorems. This dual role highlights its effectiveness as a unifying concept in the study of CSK families.

For clarity of exposition, the article is organized as follows. Section 2 reviews the concept of the VF for CSK families, emphasizing its fundamental role in the analysis to follow. Section 3 presents several properties of the FMF and illustrates how these properties interact with the $(a,b)$-transformation. In particular, we prove that if a measure belongs to the FMF, then its $(a,b)$-deformation, whether considered at the level of measures or free convolutions, also belongs to the same family. Finally, Section 4 establishes new limit theorems by means of the VF. These results concern the ${⊞}_{\left(\mathbf{t}\right)}$-convolution and extend to the setting of additive Boolean and free convolutions, thereby enriching the theoretical framework of transformations in free probability.

The implications of the present study extend well beyond the analytic characterization of CSK families. In the broader context of noncommutative probability, the $(a,b)$-deformation provides a powerful mechanism for modeling deformation and stability phenomena that naturally arise in random matrix theory and operator algebras. In random matrix theory, many central distributions of free probability such as the semicircle, Marchenko-Pastur, and free binomial laws, appear as limiting spectral distributions of large matrix ensembles. The invariance of the FMF under the $(a,b)$-deformation therefore highlights a remarkable robustness of these spectral laws under analytic perturbations, revealing a potential connection to universality phenomena in random matrices. From an operator-algebraic perspective, the $(a,b)$-deformation enriches the operational calculus on noncommutative random variables by introducing new deformation parameters that interpolate between free and Boolean convolutions. This analytic flexibility provides a unified framework for studying free convolution semigroups, noncommutative transport, and entropy-type invariants, thereby deepening the interplay between algebraic structures and probabilistic dynamics in the free setting.

2. Preliminaries

In the framework of free probability, the CSK defined by $1/(1-\vartheta s)$ is used in place of the exponential kernel $exp\left(\vartheta s\right)$. This replacement allows one to introduce a new class of families of probability measures, called CSK families, which are defined in close analogy with the classical natural exponential families (NEFs). The concept of CSK families was first studied in detail in [15,16] in the case of compactly supported probability measures. Later, the construction was extended in [17] to include probability measures whose support is bounded on one side, for instance bounded from above.

To make the setting precise, let us denote by ${\mathcal{P}}_{ba}$ the collection of (non-degenerate) probability measures with support bounded from above, and by ${\mathcal{P}}_c$ the set of probability measures with compact support. Suppose that $\eta \in {\mathcal{P}}_{ba}$. Define

$$1/{\vartheta }_{+}^{\eta }=max\{0,sup\mathrm{supp}\left(\eta \right)\}.$$

For such a measure $\eta$, one introduces the transform

$${\mathcal{M}}_{\eta }\left(\vartheta \right)=\int {\displaystyle \frac{\eta \left(ds\right)}{1-s\vartheta }}$$

which is well defined whenever $0\le \vartheta <{\vartheta }_{+}^{\eta }$.

With this definition in hand, the CSK family generated by $\eta$ is given by

$${\mathcal{K}}_{+}\left(\eta \right)=\left\{P_{\vartheta }^{\eta }\left(ds\right)={\displaystyle \frac{\eta \left(ds\right)}{{\mathcal{M}}_{\eta }\left(\vartheta \right)(1-\vartheta s)}}:0<\vartheta <{\vartheta }_{+}^{\eta }\right\}.$$

For each $\vartheta$, the measure $P_{\vartheta }^{\eta }$ is absolutely continuous with respect to $\eta$, and the collection ${\mathcal{K}}_{+}\left(\eta \right)$ describes how $\eta$ deforms under the kernel.

The mean function associated with ${\mathcal{K}}_{+}\left(\eta \right)$ is defined by

$$k_{\eta }\left(\vartheta \right)=\int s{P}_{\vartheta }^{\eta }\left(ds\right).$$

This function is bijective from the interval $(0,{\vartheta }_{+}^{\eta })$ onto the so-called mean domain, which is the open interval $(m_0^{\eta },m_{+}^{\eta })=k_{\eta }\left((0,{\vartheta }_{+}^{\eta })\right)$. This domain is referred to as the one-sided mean domain of the CSK family. The inverse function of $k_{\eta }(·)$ is denoted by ${\psi }_{\eta }(·)$. Using this inverse, one obtains the mean parameterization of the family, namely

$${\mathcal{K}}_{+}\left(\eta \right)=\{Q_u^{\eta }\left(ds\right)=P_{{\psi }_{\eta }\left(u\right)}^{\eta }\left(ds\right):m_0^{\eta }<u<m_{+}^{\eta }\}.$$

It is also useful to express the boundary points of the mean domain explicitly. Setting $B_{\eta }={\displaystyle \frac{1}{{\vartheta }_{+}^{\eta }}}$, according to [17], one has

$${\displaystyle m_0^{\eta }=\underset{\vartheta \to 0^{+}}{lim}{k}_{\eta }\left(\vartheta \right),m_{+}^{\eta }=B_{\eta }-\underset{\varpi \to B_{\eta }^{+}}{lim}1/{\mathcal{G}}_{\eta }\left(\varpi \right).}$$

The above construction corresponds to the case where the support of $\eta$ is bounded from above. A similar construction can be carried out when the support is bounded from below. In this case, the corresponding CSK family is denoted by ${\mathcal{K}}_{-}\left(\eta \right)$. With $A_{\eta }=min\{inf\mathrm{supp}\left(\eta \right),0\}$, the parameter $\vartheta$ varies in the interval $({\vartheta }_{-}^{\eta },0)$, where ${\vartheta }_{-}^{\eta }$ equals either $1/A_{\eta }$ or $-\infty$, depending on the support of $\eta$. The mean domain for this case is given by $(m_{-}^{\eta },m_0^{\eta })$, with

$$m_{-}^{\eta }=A_{\eta }-1/{\mathcal{G}}_{\eta }\left(A_{\eta }\right).$$

Finally, when $\eta$ has compact support, i.e., $\eta \in {\mathcal{P}}_c$, one can combine both constructions. In that case, the parameter $\vartheta$ ranges over the entire interval $({\vartheta }_{-}^{\eta },{\vartheta }_{+}^{\eta })$, and the resulting family is called the two-sided CSK family, denoted

$$\mathcal{K}\left(\eta \right)={\mathcal{K}}_{+}\left(\eta \right)\cup \left\{\eta \right\}\cup {\mathcal{K}}_{-}\left(\eta \right).$$

This unifies both the upper- and lower-bounded cases into a single framework.

Let $\eta \in {\mathcal{P}}_{ba}$. The VF associated with $\eta$ is introduced in [16] and is defined by

$$u\mapsto {\mathcal{V}}_{\eta }\left(u\right)=\int {(s-u)}^2{Q}_u^{\eta }\left(ds\right).$$

This function measures the variance of the distribution $Q_u^{\eta }$ around its mean u.

In situations where the measure $\eta$ does not possess a finite first moment, the classical definition of variance cannot be applied in the usual way. In such cases, all probability measures belonging to the CSK family ${\mathcal{K}}_{+}\left(\eta \right)$ automatically have infinite variance. To overcome this limitation, an alternative concept was introduced in [17]. This concept is the pseudo-variance function (PVF), denoted by ${\mathbb{V}}_{\eta }(·)$, and defined as

$${\mathbb{V}}_{\eta }\left(u\right)=u\left({\displaystyle \frac{1}{{\psi }_{\eta }\left(u\right)}}-u\right).$$

The PVF serves as a natural extension of the VF when the first moment does not exist, thereby allowing one to study the variability of CSK families in a more general framework. Moreover, when the first moment of $\eta$ does exist, that is when

$$m_0^{\eta }=\int s\eta \left(ds\right)<\infty ,$$

then the ordinary VF is well defined. In this case, the relationship between the PVF and the VF becomes explicit. In fact, it was established in [17] that

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

Thus, the PVF generalizes the classical VF while coinciding with it whenever the necessary integrability conditions are satisfied.

The following remark conclude this section by providing some useful details that reinforce the main conclusions of the article.

Remark 1.

(i) 

According to [17] measure $\eta \in {\mathcal{P}}_{ba}$ can be uniquely characterized in terms of its corresponding PVF ${\mathbb{V}}_{\eta }(·)$. To see this, let us define the auxiliary function

$$\varpi =\varpi \left(u\right)=u+{\displaystyle \frac{{\mathbb{V}}_{\eta }\left(u\right)}{u}}.$$

This function plays a central role in linking the PVF with the CST of the measure. In fact, we obtain the identity

$${\mathcal{G}}_{\eta }\left(\varpi \right)={\displaystyle \frac{u}{{\mathbb{V}}_{\eta }\left(u\right)}}.$$

When the measure η has a finite first moment $m_0^{\eta }$, this relation can be expressed in terms of the VF ${\mathcal{V}}_{\eta }(·)$. More precisely, in this case one has

$${\mathcal{G}}_{\eta }\left(\varpi \right)={\displaystyle \frac{u-m_0^{\eta }}{{\mathcal{V}}_{\eta }\left(u\right)}}.$$

(1)

This shows that both the PVF and the VF are deeply connected to the analytic properties of ${\mathcal{G}}_{\eta }(·)$. Consequently, the measure η is completely determined once the VF ${\mathcal{V}}_{\eta }(·)$ and the first moment $m_0^{\eta }$ are known. In other words, these two quantities encapsulate the essential information required to reconstruct the measure.

(ii) 

Let us consider the affine transformation of the real line given by the mapping

$$f:s\mapsto \upsilon s+\varrho ,$$

where the parameter ϱ is a real number and $\upsilon \ne 0$ is a scaling factor. Applying this transformation to the measure η, we obtain a new measure denoted by $f\left(\eta \right)$. Following [17] (Section 3.3), the effect of this transformation on the mean is straightforward: the first moment of the transformed measure is

$$m_0^{f\left(\eta \right)}=f\left(m_0^{\eta }\right)=\upsilon m_0^{\eta }+\varrho .$$

Now, let u be sufficiently close to this new mean $m_0^{f\left(\eta \right)}$. In this case, the PVF corresponding to the transformed measure satisfies the relation

$${\mathbb{V}}_{f\left(\eta \right)}\left(u\right)={\displaystyle \frac{{\upsilon }^{2}u}{u-\varrho }}{\mathbb{V}}_{\eta }\left({\displaystyle \frac{u-\varrho }{\upsilon }}\right).$$

This formula explicitly describes how the PVF changes under an affine transformation, combining both the scaling and the translation parameters.

Moreover, when the VF ${\mathcal{V}}_{\eta }(·)$ exists for the original measure, an even simpler relation holds. In fact, the VF corresponding to the transformed measure is given by

$${\mathcal{V}}_{f\left(\eta \right)}\left(u\right)={\upsilon }^2{\mathcal{V}}_{\eta }\left({\displaystyle \frac{u-\varrho }{\upsilon }}\right).$$

(2)

Thus, the effect of the affine map on the VF consists of a scaling by ${\upsilon }^2$ and a shift of the argument by ϱ, showing the structural stability of the VF under such transformations.

(iii) 

Let $\eta \in {\mathcal{P}}_{ba}$ be a probability measure with a finite first moment. One of the central analytic tools in free probability is the free cumulant transform (FCT) ${\mathcal{R}}_{\eta }$, which was first introduced in [18]. This transform is defined through its relation with the CST ${\mathcal{G}}_{\eta }$ of η, namely

$${\mathcal{R}}_{\eta }\left({\mathcal{G}}_{\eta }\left(\omega \right)\right)=\omega -1/{\mathcal{G}}_{\eta }\left(\omega \right),\omega close to\infty .$$

This identity connects the FCT to the analytic structure of ${\mathcal{G}}_{\eta }$, and it plays a crucial role in the study of free convolution.

The concept of $(a,b)$-deformation extends this framework. In [10], the $(a,b)$-deformed FCT ${\mathcal{R}}_{\eta }^{\left(\mathit{t}\right)}$, where $\mathit{t}=(a,b)$, was defined as

$${\mathcal{R}}_{\eta }^{\left(\mathit{t}\right)}\left(z\right):={\displaystyle \frac{1}{b}}\left({\mathcal{R}}_{{\tilde{U}}^{\mathit{t}}\left(\eta \right)}\left(z\right)+(b-a)m_0^{\eta }\right),$$

This formula reveals how the deformation interacts with the original FCT and the first moment $m_0^{\eta }$.

An important property of this deformed transform is its behavior under powers of $\mathit{t}$-deformed convolution ${⊞}_{\left(\mathit{t}\right)}$. For $\gamma >0$, provided that ${\eta }^{{⊞}_{\left(\mathit{t}\right)}\gamma }$ is well defined, one has

$${\mathcal{R}}_{{\eta }^{{⊞}_{\left(\mathit{t}\right)}\gamma }}^{\left(\mathit{t}\right)}\left(z\right)=\gamma {\mathcal{R}}_{\eta }^{\left(\mathit{t}\right)}\left(z\right).$$

This result mirrors the classical property of FCT under free convolution powers, but adapted to the $(a,b)$-deformed setting.

Furthermore, Theorem 1 in [11] provides an explicit description of how the VF transforms under ${\tilde{U}}^{\mathit{t}}$. More precisely, for values of u sufficiently close to the mean $m_0^{{\tilde{U}}^{\mathit{t}}\left(\eta \right)}=a{m}_0^{\eta }$, one has

$${\mathcal{V}}_{{\tilde{U}}^{\mathit{t}}\left(\eta \right)}\left(u\right)=b{\mathcal{V}}_{\eta }\left((u+(b-a)m_0^{\eta })/b\right)+\left(u-a{m}_0^{\eta }\right)\left((u+(b-a)m_0^{\eta })/b-u\right).$$

(3)

This identity highlights the precise way in which both the scaling factor b and the shift $(b-a)m_0^{\eta }$ influence the VF. In addition, Theorem 2 of [11] establishes the corresponding formula for $\mathit{t}$-deformed free convolution powers. Specifically, for u close to the mean $m_0^{{\eta }^{{⊞}_{\left(\mathit{t}\right)}\gamma }}=\gamma m_0^{\eta }$, we have

$${\mathcal{V}}_{{\eta }^{{⊞}_{\left(\mathit{t}\right)}\gamma }}\left(u\right)=\gamma {\mathcal{V}}_{\eta }(u/\gamma )+(1/\gamma -1)(u-\gamma m_0^{\eta })(u(1-b)+\gamma (b-a)m_0^{\eta }).$$

(4)

This formula generalizes the VF relation for free convolution powers by incorporating the parameters of the $(a,b)$-deformation.

3. Notes on the FMF

The CSK families having quadratic VF

$${\mathcal{V}}_{\eta }\left(u\right)=1+\alpha u+\beta u^2,\alpha \in \mathbb{R},\beta \ge -1,m_0^{\eta }=0$$

(5)

were explicitly determined in [16]. The relative laws belongs to the FMF:

$$\eta \left(dl\right)={\displaystyle \frac{\sqrt{4(1+\beta )-{(l-\alpha )}^2}}{2\pi (\beta l^2+\alpha l+1)}}{1}_{(\alpha -2\sqrt{1+\beta },\alpha +2\sqrt{1+\beta })}\left(l\right)dl+k_1{\delta }_{{\zeta }_1}+k_2{\delta }_{{\zeta }_2}.$$

(6)

We have:

(i)

if $\beta =0,{\alpha }^2>1$, then $k_1=1-1/{\alpha }^2,x_1=-1/\alpha ,k_2=0$.

(ii)

if $\beta >0$ and ${\alpha }^2>4\beta$, then $k_1=max\{0,1-{\displaystyle \frac{\left|\alpha \right|-\sqrt{{\alpha }^2-4\beta }}{2\beta \sqrt{{\alpha }^2-4\beta }}}\},k_2=0$, and ${\zeta }_1=\pm {\displaystyle \frac{\left|\alpha \right|-\sqrt{{\alpha }^2-4\beta }}{2\beta }}$ with the sign opposite to the sign of $\alpha$.

(iii)

if $-1\le \beta <0$, then

$${\zeta }_{1,2}={\displaystyle \frac{-\alpha \pm \sqrt{{\alpha }^2-4\beta }}{2\beta }},k_{1,2}=1+{\displaystyle \frac{\sqrt{{\alpha }^2-4\beta }\mp \alpha }{2\beta \sqrt{{\alpha }^2-4\beta }}}.$$

Important measures $\eta$ that have been explored in the literature are covered by this result:

(i)

The semicircle ($\mathcal{SC}$) measure if $\alpha =\beta =0$.

(ii)

The Marchenko-Pastur ($\mathcal{MP}$) measure if $\beta =0$ and $\alpha \ne 0$.

(iii)

The free Pascal ($\mathcal{FP}$) measure if $\beta >0$ and ${\alpha }^2>4\beta$.

(iv)

The free Gamma ($\mathcal{FG}$) measure if $\beta >0$ and ${\alpha }^2=4\beta$.

(v)

The free analog of hyperbolic ($\mathcal{FH}$) measure if $\beta >0$ and ${\alpha }^2<4\beta$.

(vi)

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

The FMF is a notable class of probability measures in free probability theory characterized by a quadratic VF. One remarkable property of this family is its invariance under the $(a,b)$-transformation of measures. This means that when a measure from the FMF is transformed via ${\tilde{U}}^{\mathbf{t}}(·)$, the resulting measure still belongs to the same family. More precisely we have the following result. Here, $D_q\left(\eta \right)$ denotes the measure dilation of $\eta$ by $q\ne 0$.

Theorem 1.

Assume that $\eta \in$ FMF. Then, for $\mathit{t}=(a,b)$ with $a\in \mathbb{R}$ and $b>0$, we have $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathit{t}}\left(\eta \right)\right)\in$ FMF.

Proof. 

Assume that $\eta \in$ FMF. Then,

$${\mathcal{V}}_{\eta }\left(u\right)=1+{\alpha }^{\prime }u+{\beta }^{\prime }{u}^2,{\beta }^{\prime }\ge -1,{\alpha }^{\prime }\in \mathbb{R}.$$

(7)

Combining (2), (3) and (7), for u close enough to $m_0^{D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(\eta \right)\right)}=0$, we have

$${\mathcal{V}}_{D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(\eta \right)\right)}\left(u\right)=1+{\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{b}}}u+\left({\displaystyle \frac{{\beta }^{\prime }+1}{b}}-1\right)u^2,$$

(8)

which is a VF of the form (5). Then, $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(\eta \right)\right)\in$ FMF. □

Theorem 1 establishes that the FMF is closed under the $(a,b)$-transformation of measures. This result highlights the internal symmetry and structural stability of the FMF with respect to the $(a,b)$-transformation. It further confirms that the analytic and algebraic properties defining this family, such as the form of its CST and its VF, are preserved under the combined action of scaling and translation encoded by the parameters a and b. Next, we present several examples of measures that illustrate the significance of Theorem 1.

Corollary 1.

Let $sb={\displaystyle \frac{1}{2}}{\delta }_{-1}+{\displaystyle \frac{1}{2}}{\delta }_1$ be the symmetric Bernoulli ($\mathcal{SB}$) law. Then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathit{t}}\left(sb\right)\right)=sb$.

Proof. 

The VF of $\mathcal{K}\left(sb\right)$ is provided by (7) for ${\beta }^{\prime }=-1$ and ${\alpha }^{\prime }=0$. Using (8), we obtain

$${\mathcal{V}}_{D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(sb\right)\right)}\left(u\right)=1-u^2={\mathcal{V}}_{sb}\left(u\right),u\mathrm{close} \mathrm{to}{m}_0^{D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(sb\right)\right)}=0.$$

The proof is concluded by the use of (1). □

Corollary 2.

Consider the $\mathcal{SC}$ measure

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

(9)

Then, $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathit{t}}\left(sc\right)\right)$ is

(i) 

a $\mathcal{SC}$ measure if $b=1$.

(ii) 

a $\mathcal{FH}$ measure with $\alpha =0$ and $\beta ={\displaystyle \frac{1}{b}}-1>0$ if $0<b<1$.

(iii) 

a $\mathcal{FB}$ measure with $\alpha =0$ and $-1\le \beta ={\displaystyle \frac{1}{b}}-1<0$ if $1<b<+\infty$.

Proof. 

For $\mathcal{K}\left(sc\right)$, the VF is provided by (7) for ${\alpha }^{\prime }={\beta }^{\prime }=0$. From (8), we have that

$${\mathcal{V}}_{D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(sc\right)\right)}\left(u\right)=1+(1/b-1)u^2,u\mathrm{close} \mathrm{to}{m}_0^{D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(sc\right)\right)}=0.$$

(10)

Identifying (5) and (10), we get

(i)

If $b=1$, then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(sc\right)\right)=sc$.

(ii)

If $0<b<1$, then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(sc\right)\right)$ is a $\mathcal{FH}$ measure (6) with $\alpha =0$ and $\beta ={\displaystyle \frac{1}{b}}-1>0$.

(iii)

If $1<b<+\infty$, then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(sc\right)\right)$ is a $\mathcal{FB}$ measure (6) with $\alpha =0$ and $-1\le \beta ={\displaystyle \frac{1}{b}}-1<0$.

□

Corollary 3.

For ${\alpha }^{\prime }\ne 0$ and ${\beta }^{\prime }=0$, consider the $\mathcal{MP}$ measure

$$mp\left(dl\right)={\displaystyle \frac{\sqrt{4-{({\alpha }^{\prime }-l)}^2}}{2\pi (l{\alpha }^{\prime }+1)}}{1}_{({\alpha }^{\prime }-2,{\alpha }^{\prime }+2)}\left(l\right)dl+{\left(1-{\displaystyle \frac{1}{{\alpha }^{\prime 2}}}\right)}^{+}{\delta }_{{\displaystyle \frac{-1}{{\alpha }^{\prime }}}}.$$

(11)

Then, $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathit{t}}\left(mp\right)\right)$ is

(i) 

a $\mathcal{MP}$ measure with $\alpha ={\alpha }^{\prime }$ and $\beta ={\beta }^{\prime }=0$ if $b=1$.

(ii) 

a $\mathcal{FP}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{b}$ and $\beta =1/b-1>0$ if $b\in (0,1)$ and ${\alpha }^{\prime 2}/b>4(1/b-1)$.

(iii) 

a $\mathcal{FG}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{b}$ and $\beta =1/b-1>0$ if $b\in (0,1)$ and ${\alpha }^{\prime 2}/b=4(1/b-1)$.

(iv) 

a $\mathcal{FH}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{b}$ and $\beta =1/b-1>0$ if $b\in (0,1)$ and ${\alpha }^{\prime 2}/b<4(1/b-1)$.

(v) 

a $\mathcal{FB}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{b}$ and $-1\le \beta =1/b-1<0$ if $b\in (1,+\infty )$.

Proof. 

The VF of $\mathcal{K}\left(mp\right)$ is provided by (7) for ${\beta }^{\prime }=0$ and ${\alpha }^{\prime }\ne 0$. Relation (8) gives

$${\mathcal{V}}_{D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(mp\right)\right)}\left(u\right)=1+{\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{b}}}u+(1/b-1)u^2,u\mathrm{close} \mathrm{to}{m}_0^{D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(mp\right)\right)}=0.$$

(12)

Identifying (5) and (12), we get

(i)

If $b=1$, then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(mp\right)\right)=mp$.

(ii)

If $0<b<1$ and ${\displaystyle \frac{{\alpha }^{\prime 2}}{b}}>4({\displaystyle \frac{1}{b}}-1)$, then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(mp\right)\right)$ is a $\mathcal{FP}$ measure (6) with $\alpha ={\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{b}}}$ and $\beta ={\displaystyle \frac{1}{b}}-1>0$.

(iii)

If $0<b<1$ and ${\displaystyle \frac{{\alpha }^{\prime 2}}{b}}=4({\displaystyle \frac{1}{b}}-1)$, then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(mp\right)\right)$ is a $\mathcal{FG}$ measure (6) with $\alpha ={\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{b}}}$ and $\beta ={\displaystyle \frac{1}{b}}-1>0$.

(iv)

If $0<b<1$ and ${\displaystyle \frac{{\alpha }^{\prime 2}}{b}}<4({\displaystyle \frac{1}{b}}-1)$, then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(mp\right)\right)$ is a $\mathcal{FH}$ measure (6) with $\alpha ={\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{b}}}$ and $\beta ={\displaystyle \frac{1}{b}}-1>0$.

(v)

If $1<b<+\infty$, then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(mp\right)\right)$ is a $\mathcal{FB}$ measure (6) with $\alpha ={\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{b}}}$ and $-1\le \beta ={\displaystyle \frac{1}{b}}-1<0$.

□

Corollary 4.

For ${\alpha }^{\prime }\ne 0$ and ${\beta }^{\prime }={\displaystyle \frac{{\alpha }^{\prime 2}}{4}}$, consider the $\mathcal{FG}$ measure

$$fg\left(dl\right)={\displaystyle \frac{\sqrt{4(1+{\beta }^{\prime })-{({\alpha }^{\prime }-l)}^2}}{2\pi ({\beta }^{\prime }{l}^2+{\alpha }^{\prime }l+1)}}{1}_{({\alpha }^{\prime }-2\sqrt{{\beta }^{\prime }+1},{\alpha }^{\prime }+2\sqrt{{\beta }^{\prime }+1})}\left(l\right)dl.$$

(13)

Then, $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathit{t}}\left(fg\right)\right)$ is

(i) 

a $\mathcal{MP}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{b}$ and $\beta =0$, if $b=1+{\alpha }^{\prime 2}/4$.

(ii) 

a $\mathcal{FG}$ measure with $\alpha ={\alpha }^{\prime }$ and $\beta ={\beta }^{\prime }$ if $b=1$.

(iii) 

a $\mathcal{FH}$ measure with $\alpha ={\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{b}}}$ and $\beta ={\displaystyle \frac{{\beta }^{\prime }+1}{b}}-1={\displaystyle \frac{{\alpha }^{\prime 2}/4+1}{b}}-1>0$ if $b\in (0,1)$.

(iv) 

a $\mathcal{FP}$ measure with $\alpha ={\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{b}}}$ and $\beta ={\displaystyle \frac{{\beta }^{\prime }+1}{b}}-1={\displaystyle \frac{{\alpha }^{\prime 2}/4+1}{b}}-1>0$ if $b\in (1,+\infty )$ such that ${\alpha }^{\prime 2}/b+4(1/b-1)>0$.

(v) 

a $\mathcal{FB}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{b}$ and $\beta ={\displaystyle \frac{{\beta }^{\prime }+1}{b}}-1={\displaystyle \frac{{\alpha }^{\prime 2}/4+1}{b}}-1$ if $b\in (1,+\infty )$, such that $-1\le {\displaystyle \frac{{\alpha }^{\prime 2}/4+1}{b}}-1<0$.

Proof. 

The VF of $\mathcal{K}\left(fg\right)$ provided by (7) for ${\alpha }^{\prime }\ne 0$ and ${\beta }^{\prime }={\alpha }^{\prime 2}/4$. From (8), we get

$${\mathcal{V}}_{D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(fg\right)\right)}\left(u\right)=1+{\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{b}}}u+\left({\displaystyle \frac{1+{\beta }^{\prime }}{b}}-1\right)u^2=1+{\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{b}}}u+\left({\displaystyle \frac{{\alpha }^{\prime 2}/4+1}{b}}-1\right)u^2,$$

(14)

for u close to $m_0^{D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(fg\right)\right)}=0$. Identifying (14) and (5), we get

(i)

If $b=1+{\alpha }^{\prime 2}/4$, then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(fg\right)\right)$ is a $\mathcal{MP}$ measure (6) with $\alpha ={\alpha }^{\prime }/\sqrt{b}$ and $\beta =0$.

(ii)

If $b=1$, then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(fg\right)\right)=fg$.

(iii)

If $0<b<1$, then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(fg\right)\right)$ is a $\mathcal{FH}$ measure (6) with $\alpha ={\alpha }^{\prime }/\sqrt{b}$ and $\beta ={\displaystyle \frac{1+{\beta }^{\prime }}{b}}-1={\displaystyle \frac{{\alpha }^{\prime 2}/4+1}{b}}-1>0$.

(iv)

If $b\in (1,+\infty )$ such that ${\alpha }^{\prime 2}/b+4(1/b-1)>0$, then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(fg\right)\right)$ is a $\mathcal{FP}$ measure (6) with $\alpha ={\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{b}}}$ and $\beta ={\displaystyle \frac{{\beta }^{\prime }+1}{b}}-1={\displaystyle \frac{{\alpha }^{\prime 2}/4+1}{b}}-1>0$.

(v)

If $b\in (1,+\infty )$, such that $-1\le {\displaystyle \frac{{\alpha }^{\prime 2}/4+1}{b}}-1<0$, then $D_{1/\sqrt{b}}\left({\tilde{U}}^{\mathbf{t}}\left(fg\right)\right)$ is a $\mathcal{FB}$ measure (6) with $\alpha ={\alpha }^{\prime }/\sqrt{b}$ and $\beta ={\displaystyle \frac{{\beta }^{\prime }+1}{b}}-1={\displaystyle \frac{{\alpha }^{\prime 2}/4+1}{b}}-1$.

□

A notable property of the $(a,b)$-transformation of measures is its ability, for suitable choices of parameters, to map the $\mathcal{FG}$ measure onto the $\mathcal{MP}$ measure. This transformation thus establishes a direct and elegant connection between two fundamental distributions in free probability theory. Such a link cannot be achieved through the t-transformation alone, which highlights the broader flexibility and stronger unifying capacity of the $(a,b)$-transformation. In this sense, it provides a more comprehensive framework for relating and deforming key probabilistic structures within the free setting.

The FMF’s invariance property under ${⊞}_{\left(\mathbf{t}\right)}$-convolution power is now stated and shown.

Theorem 2.

Assume that $\eta \in$ FMF. Then, for $\gamma >0$ so that ${\eta }^{{⊞}_{\left(\mathit{t}\right)}\gamma }$ is defined, we have $D_{{\displaystyle \frac{1}{\sqrt{\gamma }}}}\left({\eta }^{{⊞}_{\left(\mathit{t}\right)}\gamma }\right)\in$ FMF.

Proof. 

Assume that $\eta \in$ FMF. Using (7), (4) and (2), for u close enough to $m_0^{D_{1/\sqrt{\gamma }}\left({\eta }^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)}=0$, one has

$${\mathcal{V}}_{D_{1/\sqrt{\gamma }}\left({\eta }^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)}\left(u\right)=1+{\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{\gamma }}}u+\left({\displaystyle \frac{(1-b)(1-\gamma )+{\beta }^{\prime }}{\gamma }}\right)u^2,$$

(15)

which is a VF of the form (5). Then, $D_{1/\sqrt{\gamma }}\left({\eta }^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)\in$ FMF. □

Theorem 2 demonstrates that the FMF is closed under the ${⊞}_{\left(\mathbf{t}\right)}$-convolution power. This closure property reveals a fundamental symmetry and invariance within the family, showing that the deformation acts internally rather than producing measures outside the class. As a result, Theorem 2 provides a significant contribution to the theory of free convolutions by identifying one of the rare non-trivial families that remain structurally invariant under the $(a,b)$-transformation. Next, the significance of Theorem 2 is emphasized across a few specific measures.

Corollary 5.

Consider $sb={\displaystyle \frac{1}{2}}{\delta }_{-1}+{\displaystyle \frac{1}{2}}{\delta }_1$. Then, $D_{1/\sqrt{\gamma }}\left(s{b}^{{⊞}_{\left(\mathit{t}\right)}\gamma }\right)$ is

(i) 

a $\mathcal{SB}$ measure if $\gamma =1$.

(ii) 

a $\mathcal{SC}$ measure if $\gamma \in (1,+\infty )$ and $b={\displaystyle \frac{1}{1-1/\gamma }}$.

(iii) 

a $\mathcal{FH}$ measure with $\alpha =0$ and $\beta ={\displaystyle \frac{-b}{\gamma }}+b-1$ if $\gamma \in (1,+\infty )$ and $b>{\displaystyle \frac{1}{1-1/\gamma }}$.

(iv) 

a $\mathcal{FB}$ measure with $\alpha =0$ and $\beta ={\displaystyle \frac{-b}{\gamma }}+b-1$ if $\gamma \in (1,+\infty )$ and $b<{\displaystyle \frac{1}{1-1/\gamma }}$.

Proof. 

For $\mathcal{K}\left(sb\right)$, the VF is given by (7) for ${\alpha }^{\prime }=0$ and ${\beta }^{\prime }=-1$. Equation (15) gives

$${\mathcal{V}}_{D_{1/\sqrt{\gamma }}\left(s{b}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)}\left(u\right)=1+\left({\displaystyle \frac{-b}{\gamma }}+b-1\right)u^2,u\mathrm{close} \mathrm{to}{m}_0^{D_{1/\sqrt{\gamma }}\left(s{b}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)}=0.$$

(16)

Comparing (16) to (5), we obtain:

(i)

If $\gamma =1$, then $D_{1/\sqrt{\gamma }}\left(s{b}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)=sb$.

(ii)

If $\gamma \in (1,+\infty )$ and $b={\displaystyle \frac{1}{1-1/\gamma }}$, then $D_{1/\sqrt{\gamma }}\left(s{b}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)$ is a $\mathcal{SC}$ measure (6) with $\alpha =\beta =0$.

(iii)

If $\gamma \in (1,+\infty )$ and $b>{\displaystyle \frac{1}{1-1/\gamma }}$, then $D_{1/\sqrt{\gamma }}\left(s{b}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)$ a $\mathcal{FH}$ measure (6) with $\alpha =0$ and $\beta ={\displaystyle \frac{-b}{\gamma }}+b-1$.

(iv)

If $\gamma \in (1,+\infty )$ and $b<{\displaystyle \frac{1}{1-1/\gamma }}$, then $D_{1/\sqrt{\gamma }}\left(s{b}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)$ a $\mathcal{FB}$ measure (6) with $\alpha =0$ and $\beta ={\displaystyle \frac{-b}{\gamma }}+b-1$.

□

Corollary 6.

Consider the $\mathcal{SC}$ measure (9). Then, $D_{1/\sqrt{\gamma }}\left(s{c}^{{⊞}_{\left(\mathit{t}\right)}\gamma }\right)$ is

(i) 

a $\mathcal{SC}$ measure if $\gamma =1$ or $b=1$.

(ii) 

a $\mathcal{FH}$ measure with $\alpha =0$ and $\beta ={\displaystyle \frac{(1-b)(1-\gamma )}{\gamma }}$ if $\gamma \in (1,+\infty )$ and $b\in (1,+\infty )$ (or $\gamma \in (0,1)$ and $b\in (0,1)$).

(iii) 

a $\mathcal{FB}$ measure with $\alpha =0$ and $\beta ={\displaystyle \frac{(1-b)(1-\gamma )}{\gamma }}$ if $\gamma \in (1,+\infty )$ and $b\in (0,1)$ (or $\gamma \in (0,1)$ and $b\in (1,+\infty )$ such that ${\displaystyle \frac{(1-b)(1-\gamma )}{\gamma }}\ge -1$).

Proof. 

For the $\mathcal{SC}$ measure (9), then relation (15) reduces to

$${\mathcal{V}}_{D_{1/\sqrt{\gamma }}\left(s{c}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)}\left(u\right)=1+{\displaystyle \frac{(1-b)(1-\gamma )}{\gamma }}{u}^2,u\mathrm{close} \mathrm{to}{m}_0^{D_{1/\sqrt{\gamma }}\left(s{c}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)}=0.$$

(17)

Comparing (17) to (5), we obtain:

(i)

If $\gamma =1$ or $b=1$, then $D_{1/\sqrt{\gamma }}\left(s{c}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)=sc.$

(ii)

If $\gamma \in (1,+\infty )$ and $b\in (1,+\infty )$ (or $\gamma \in (0,1)$ and $b\in (0,1)$), then $D_{1/\sqrt{\gamma }}\left(s{c}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)$ is a $\mathcal{FH}$ measure (6) with $\alpha =0$ and $\beta ={\displaystyle \frac{(1-b)(1-\gamma )}{\gamma }}$.

(iii)

If $\gamma \in (1,+\infty )$ and $b\in (0,1)$ (or $\gamma \in (0,1)$ and $b\in (1,+\infty )$ such that ${\displaystyle \frac{(1-b)(1-\gamma )}{\gamma }}\ge -1$), then $D_{1/\sqrt{\gamma }}\left(s{c}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)$ is a $\mathcal{FB}$ measure (6) with $\alpha =0$ and $\beta ={\displaystyle \frac{(1-b)(1-\gamma )}{\gamma }}$.

□

Corollary 7.

Consider the $\mathcal{MP}$ measure (11). Then, $D_{1/\sqrt{\gamma }}\left(m{p}^{{⊞}_{\left(\mathit{t}\right)}\gamma }\right)$ is

(i) 

a $\mathcal{MP}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\beta }^{\prime }=0$ if $\gamma =1$ or $b=1$.

(ii) 

a $\mathcal{FP}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\displaystyle \frac{(1-\gamma )(1-b)}{\gamma }}$ if $\gamma \in (0,1)$ and $b\in (0,1)$ (or $\gamma \in (1,+\infty )$ and $b\in (1,+\infty )$) such that ${\alpha }^{\prime 2}/\gamma >4{\displaystyle \frac{(1-\gamma )(1-b)}{\gamma }}$.

(iii) 

a $\mathcal{FG}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\displaystyle \frac{(1-\gamma )(1-b)}{\gamma }}$ if $\gamma \in (0,1)$ and $b\in (0,1)$ (or $\gamma \in (1,+\infty )$ and $b\in (1,+\infty )$) such that ${\alpha }^{\prime 2}/\gamma =4{\displaystyle \frac{(1-\gamma )(1-b)}{\gamma }}$.

(iv) 

a $\mathcal{FH}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\displaystyle \frac{(1-\gamma )(1-b)}{\gamma }}$ if $\gamma \in (0,1)$ and $b\in (0,1)$ (or $\gamma \in (1,+\infty )$ and $b\in (1,+\infty )$) such that ${\alpha }^{\prime 2}/\gamma <4{\displaystyle \frac{(1-\gamma )(1-b)}{\gamma }}$.

(v) 

a $\mathcal{FB}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\displaystyle \frac{(1-\gamma )(1-b)}{\gamma }}$ if $\gamma \in (1,+\infty )$ and $b\in (0,1)$ (or $\gamma \in (0,1)$ and $b\in (1,+\infty )$ such that $-1\le \beta ={\displaystyle \frac{(1-b)(1-\gamma )}{\gamma }}<0$).

Proof. 

For the $\mathcal{MP}$ measure (11), relation (15) reduces to

$${\mathcal{V}}_{D_{1/\sqrt{\gamma }}\left(m{p}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\gamma }\right)}\left(u\right)=1+{\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{\gamma }}}u+{\displaystyle \frac{(1-b)(1-\gamma )}{\gamma }}{u}^2,u\mathrm{close} \mathrm{to}{m}_0^{D_{1/\sqrt{\gamma }}\left(m{p}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\gamma }\right)}=0.$$

(18)

Comparing (18) to (5), we obtain:

(i)

If $\gamma =1$ or $b=1$, then $D_{1/\sqrt{\gamma }}\left(m{p}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)=mp$.

(ii)

If $\gamma \in (0,1)$ and $b\in (0,1)$ (or $\gamma \in (1,+\infty )$ and $b\in (1,+\infty )$) such that ${\alpha }^{\prime 2}>4(1-\gamma )(1-b)$, then, $D_{1/\sqrt{\gamma }}\left(m{p}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)$ is a $\mathcal{FP}$ measure (6) with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\displaystyle \frac{(1-\gamma )(1-b)}{\gamma }}$.

(iii)

If $\gamma \in (0,1)$ and $b\in (0,1)$ (or $\gamma \in (1,+\infty )$ and $b\in (1,+\infty )$) such that ${\alpha }^{\prime 2}=4(1-\gamma )(1-b)$, then $D_{1/\sqrt{\gamma }}\left(m{p}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)$ is a $\mathcal{FG}$ measure (6) with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\displaystyle \frac{(1-\gamma )(1-b)}{\gamma }}$.

(iv)

If $\gamma \in (0,1)$ and $b\in (0,1)$ (or $\gamma \in (1,+\infty )$ and $b\in (1,+\infty )$) such that ${\alpha }^{\prime 2}<4(1-\gamma )(1-b)$, then $D_{1/\sqrt{\gamma }}\left(m{p}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)$ is a $\mathcal{FH}$ measure (6) with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\displaystyle \frac{(1-\gamma )(1-b)}{\gamma }}$.

(v)

If $\gamma \in (1,+\infty )$ and $b\in (0,1)$ (or $\gamma \in (0,1)$ and $b\in (1,+\infty )$ such that $-1\le \beta ={\displaystyle \frac{(1-b)(1-\gamma )}{\gamma }}<0$), then $D_{1/\sqrt{\gamma }}\left(m{p}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)$ is a $\mathcal{FB}$ measure (6) with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\displaystyle \frac{(1-\gamma )(1-b)}{\gamma }}$.

□

Corollary 8.

Consider the $\mathcal{FG}$ measure (13). Then, $D_{1/\sqrt{\gamma }}\left(f{g}^{{⊞}_{\left(\mathit{t}\right)}\gamma }\right)$ is

(i) 

a $\mathcal{MP}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta =0$, if ${\alpha }^{\prime 2}/4+(1-b)(1-\gamma )=0$.

(ii) 

a $\mathcal{FG}$ measure with $\alpha ={\alpha }^{\prime }$ and $\beta ={\beta }^{\prime }$ if $\gamma =1$ or $b=1$.

(iii) 

a $\mathcal{FH}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\displaystyle \frac{{\alpha }^{\prime 2}/4+(1-b)(1-\gamma )}{\gamma }}$ if $\gamma \in (0,1)$ and $b\in (0,1)$ (or $\gamma \in (1,+\infty )$ and $b\in (1,+\infty )$).

(iv) 

a $\mathcal{FP}$ measure with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\displaystyle \frac{{\alpha }^{\prime 2}/4+(1-b)(1-\gamma )}{\gamma }}$ if $\gamma \in (0,1)$ and $b\in (1,+\infty )$ (or $\gamma \in (1,+\infty )$ and $b\in (0,1)$) such that ${\alpha }^{\prime 2}>4(b-1)(1-\gamma )$.

(v) 

a $\mathcal{FB}$ measure with $\alpha ={\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{\gamma }}}$ and $\beta ={\displaystyle \frac{\frac{{\alpha }^{\prime 2}}{4}+(1-\gamma )(1-b)}{\gamma }}$ if $\gamma >1$ and $0<b<1$ (or $0<\gamma <1$ and $b>1$) such that $-1\le \beta ={\displaystyle \frac{\frac{{\alpha }^{\prime 2}}{4}+(1-\gamma )(1-b)}{\gamma }}<0$.

Proof. 

For the $\mathcal{FG}$ measure (13), relation (15) reduces to

$${\mathcal{V}}_{D_{1/\sqrt{\gamma }}\left(f{g}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)}\left(u\right)=1+{\displaystyle \frac{{\alpha }^{\prime }}{\sqrt{\gamma }}}u+\left({\displaystyle \frac{{\alpha }^{\prime 2}/4+(1-b)(1-\gamma )}{\gamma }}\right)u^2,$$

(19)

for u close to $m_0^{D_{1/\sqrt{\gamma }}\left(f{g}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)}=0$. Comparing (19) to (5), we obtain:

(i)

If ${\alpha }^{\prime 2}/4+(1-b)(1-\gamma )=0$, then $D_{1/\sqrt{\gamma }}\left(f{g}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)$ is a $\mathcal{MP}$ measure (6) with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta =0$.

(ii)

If $\gamma =1$ or $b=1$, then $D_{1/\sqrt{\gamma }}\left(f{g}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)=fg$.

(iii)

If $\gamma \in (0,1)$ and $b\in (0,1)$ (or $\gamma \in (1,+\infty )$ and $b\in (1,+\infty )$), then $D_{1/\sqrt{\gamma }}\left(f{g}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)$ is a $\mathcal{FH}$ measure (6) with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\displaystyle \frac{{\alpha }^{\prime 2}/4+(1-b)(1-\gamma )}{\gamma }}$.

(iv)

If $\gamma \in (0,1)$ and $b\in (1,+\infty )$ (or $\gamma \in (1,+\infty )$ and $b\in (0,1)$) such that ${\alpha }^{\prime 2}>4(b-1)(1-\gamma )$, then $D_{1/\sqrt{\gamma }}\left(f{g}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)$ is a $\mathcal{FP}$ measure (6) with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\displaystyle \frac{{\alpha }^{\prime 2}/4+(1-b)(1-\gamma )}{\gamma }}$.

(v)

If $\gamma >1$ and $0<b<1$ (or $0<\gamma <1$ and $b>1$) such that $-1\le \beta ={\displaystyle \frac{\frac{{\alpha }^{\prime 2}}{4}+(1-\gamma )(1-b)}{\gamma }}<0$, then $D_{1/\sqrt{\gamma }}\left(f{g}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }\right)$ is a $\mathcal{FB}$ measure (6) with $\alpha ={\alpha }^{\prime }/\sqrt{\gamma }$ and $\beta ={\displaystyle \frac{{\alpha }^{\prime 2}/4+(1-b)(1-\gamma )}{\gamma }}$.

□

A remarkable aspect of the ${⊞}_{\left(\mathbf{t}\right)}$-convolution lies in its capacity, for appropriate parameter values, to map the $\mathcal{FG}$ measure onto the $\mathcal{MP}$ measure. This property reveals a direct and elegant correspondence between two cornerstone distributions in free probability theory. Unlike the $\begin{array}{|c|}\hline t\\ \hline\end{array}$-convolution, which cannot realize such a connection, the ${⊞}_{\left(\mathbf{t}\right)}$-convolution exhibits a greater degree of structural versatility. Consequently, it serves as a powerful and unifying tool for describing, relating, and deforming fundamental probabilistic models within the framework of free probability.

The invariance property established in this study is of fundamental importance because it ensures the stability of the FMF under the $(a,b)$-transformation. In particular, the essential structural characteristics of the FMF, such as its moments, cumulants, and VFs, remain preserved when the transformation is applied. This preservation implies that the transformation acts internally within the family, allowing one to construct new probability measures from existing ones without altering the overall algebraic or probabilistic structure. From a practical perspective, this property considerably simplifies analytical and computational procedures in free probability. Researchers can manipulate and deform measures within a single unified framework while maintaining control over their main statistical and functional features. Furthermore, the invariance of the FMF under the $(a,b)$-transformation reveals a deep and elegant symmetry inherent in this family. It indicates that the algebraic, combinatorial, and functional properties of the FMF are naturally compatible with the deformation structure induced by the parameters $(a,b)$. Overall, the closure of the FMF under the $(a,b)$-transformation represents a key structural feature with both theoretical and applied implications. It not only enriches the understanding of symmetries within free probabilistic frameworks but also provides a useful analytical tool for applications in random matrix theory, operator algebras, and broader areas of non-commutative probability.

4. Novel Limiting Theorems for the ${⊞}_{\left(\mathbf{t}\right)}$-Convolution

We establish several new limiting theorems for the convolution ${⊞}_{\left(\mathbf{t}\right)}$, which incorporates both Boolean and free additive convolutions. An important observation is that, when the parameters are chosen as $a=b=1$, the operation ${⊞}_{\left(\mathbf{t}\right)}$ reduces to the standard free additive convolution ⊞. On the other hand, for $\eta \in \mathcal{P}$, the Boolean additive convolution power ${\eta }^{\uplus t}$ can be interpreted as a particular case of the $(a,b)$-transformation of measures, namely when $a=b=t>0$.

In this framework, Belinschi and Nica [19] introduced, for each $r\ge 0$, a new transformation of measures, denoted by ${\mathbb{B}}_r$. This transformation acts on the space of probability measures $\mathcal{P}$ according to

$${\mathbb{B}}_r:\mathcal{P}\to \mathcal{P},\eta \mapsto {\mathbb{B}}_r\left(\eta \right)={\left({\eta }^{⊞(1+r)}\right)}^{\uplus {\displaystyle \frac{1}{1+r}}}.$$

It was further shown that the special case ${\mathbb{B}}_1$ coincides with the celebrated Bercovici–Pata bijection, which plays a fundamental role in the study of infinite divisibility in free probability.

Let us now consider a measure $\eta \in {\mathcal{P}}_{ba}$ with finite first moment. From [17], we recall that for $k>0$, whenever the free convolution power ${\eta }^{⊞k}$ is well defined, the mean and VF satisfy the relations

$$m_0^{{\eta }^{⊞k}}=k{m}_0^{\eta }$$

and

$${\mathcal{V}}_{{\eta }^{⊞k}}\left(u\right)=k{\mathcal{V}}_{\eta }(u/k),u\mathrm{close} \mathrm{enough} \mathrm{to}k{m}_0^{\eta }.$$

(20)

Similarly, for $r>0$, the Boolean additive convolution power satisfies the following identities (see [20]):

$$m_0^{{\eta }^{\uplus r}}=r{m}_0^{\eta }$$

and

$${\mathcal{V}}_{{\eta }^{\uplus r}}\left(u\right)=r{\mathcal{V}}_{\eta }(u/r)+u(u-r{m}_0^{\eta })(-1+1/r),u\mathrm{close} \mathrm{enough} \mathrm{to}r{m}_0^{\eta }.$$

(21)

Moreover, the ${\mathbb{B}}_r$-transformation preserves the mean of the measure. More precisely, one has

$$m_0^{{\mathbb{B}}_r\left(\eta \right)}=m_0^{\eta }=m_0^{{\mathbb{B}}_1^{-1}\left(\eta \right)}.$$

The corresponding VFs are modified in a simple but important way: for u close enough to $m_0^{\eta }$

$${\mathcal{V}}_{{\mathbb{B}}_r\left(\eta \right)}\left(u\right)={\mathcal{V}}_{\eta }\left(u\right)+ru(u-m_0^{\eta }),$$

(22)

and

$${\mathcal{V}}_{{\mathbb{B}}_1^{-1}\left(\eta \right)}\left(u\right)={\mathcal{V}}_{\eta }\left(u\right)-u(u-m_0^{\eta }).$$

(23)

Next, the primary findings of this section are then stated and demonstrated.

Theorem 3.

Let $\eta \in {\mathcal{P}}_c$.

(i) 

For $\gamma >0$ and $b\in (0,1]$ so that ${\left({\eta }^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }$ is defined, we have

$${\left({\eta }^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }\stackrel{\gamma \to +\infty }{\to }{D}_b{\left(\varphi \left({\tilde{U}}^{\mathit{t}}\left(\eta \right)\right)\right)}^{⊞\frac{1}{b}}in distribution,$$

(24)

with $\varphi \left(x\right)=(x+(b-a)m_0^{\eta })/b$.

(ii) 

For $\gamma >0$ and $b\in [1,+\infty )$ so that ${\left({\eta }^{⊞\frac{1}{\gamma }}\right)}^{{⊞}_{\left(\mathit{t}\right)}\gamma }$ is defined, we have

$${\left({\eta }^{⊞\frac{1}{\gamma }}\right)}^{{⊞}_{\left(\mathit{t}\right)}\gamma }\stackrel{\gamma \to +\infty }{\to }{\left({\tilde{U}}^{\mathit{t}}\right)}^{-1}\left({\varphi }^{-1}\left(D_{1/b}\left({\eta }^{⊞b}\right)\right)\right)in distribution,$$

(25)

with ${\varphi }^{-1}\left(x\right)=bx-(b-a)m_0^{\rho }$.

(iii) 

For $\gamma >0$ and $b\in (0,1]$ so that ${\left({\eta }^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{\uplus \gamma }$ is defined, we have

$${\left({\eta }^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{\uplus \gamma }\stackrel{\gamma \to +\infty }{\to }{\mathbb{B}}_1^{-1}\left(D_b{\left(\varphi \left({\tilde{U}}^{\mathit{t}}\left(\eta \right)\right)\right)}^{⊞\frac{1}{b}}\right),in distribution.$$

(26)

(iv) 

For $\gamma >0$ and $b\in [1,+\infty )$ so that ${\left({\eta }^{\uplus \frac{1}{\gamma }}\right)}^{{⊞}_{\left(\mathit{t}\right)}\gamma }$ is defined, we have

$${\left({\eta }^{\uplus \frac{1}{\gamma }}\right)}^{{⊞}_{\left(\mathit{t}\right)}\gamma }\stackrel{\gamma \to +\infty }{\to }{\mathbb{B}}_1\left({\tilde{U}}^{{\mathit{t}}^{-1}}\left({\varphi }^{-1}\left(D_{1/b}\left({\eta }^{⊞b}\right)\right)\right)\right)in distribution,$$

(27)

Proof. 

(i) We have that

$$m_0^{{\left({\eta }^{{⊞}_{\left(\mathbf{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }}=m_0^{\eta }=m_0^{D_b{\left(\varphi \left({\tilde{U}}^{\mathbf{t}}\left(\eta \right)\right)\right)}^{⊞\frac{1}{b}}}.$$

So $\epsilon >0$ exists so that ${\mathcal{V}}_{{\left({\eta }^{{⊞}_{\left(\mathbf{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }}(·)$ and ${\mathcal{V}}_{D_b{\left(\varphi \left({\tilde{U}}^{\mathbf{t}}\left(\eta \right)\right)\right)}^{⊞\frac{1}{b}}}(·)$ are defined on $(m_0^{\rho },m_0^{\rho }+\epsilon )$.

Using (20) and (4), for $u\in (m_0^{\eta },m_0^{\eta }+\epsilon )$, one has

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{{\left({\eta }^{{⊞}_{\left(\mathbf{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }}\left(u\right)& =& \gamma {\mathcal{V}}_{{\eta }^{{⊞}_{\left(\mathbf{t}\right)}\frac{1}{\gamma }}}(u/\gamma )\hfill \\ & =& \gamma \left[{\mathcal{V}}_{\eta }\left(u\right)/\gamma +(\gamma -1)(u/\gamma -m_0^{\eta }/\gamma )(u(1-b)/\gamma +(b-a)m_0^{\eta }/\gamma )\right]\hfill \\ & =& {\mathcal{V}}_{\eta }\left(u\right)+(1-1/\gamma )(u-m_0^{\eta })(u(1-b)+(b-a)m_0^{\eta })\hfill \\ & \stackrel{\gamma \to +\infty }{\to }& {\mathcal{V}}_{\eta }\left(u\right)+(u-m_0^{\eta })(u(1-b)+(b-a)m_0^{\eta }).\hfill \end{array}$$

(28)

On the other hand, with $\varphi \left(x\right)=(x+(b-a)m_0^{\eta })/b$, we have

$${\mathcal{V}}_{D_b{\left(\varphi \left({\tilde{U}}^{\mathbf{t}}\left(\eta \right)\right)\right)}^{⊞{\displaystyle \frac{1}{b}}}}\left(u\right)=b{\mathcal{V}}_{\varphi \left({\tilde{U}}^{\mathbf{t}}\left(\eta \right)\right)}\left(u\right)={\mathcal{V}}_{\eta }\left(u\right)+(u-m_0^{\eta })(u(1-b)+(b-a)m_0^{\eta }).$$

(29)

Equations (28) and (29) gives

$${\mathcal{V}}_{{\left({\eta }^{{⊞}_{\left(\mathbf{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }}\left(u\right)\stackrel{\gamma \to +\infty }{\to }{\mathcal{V}}_{D_b{\left(\varphi \left({\tilde{U}}^{\mathbf{t}}\left(\eta \right)\right)\right)}^{⊞\frac{1}{b}}}\left(u\right).$$

This, with [16] (Proposition 4.2) complete the demonstration of (24).

It is worth emphasizing that the condition $b\in (0,1]$ is essential. This assumption guarantees that the measure $D_b{\left(\varphi \left({\tilde{U}}^{\mathbf{t}}\left(\eta \right)\right)\right)}^{⊞{\displaystyle \frac{1}{b}}}$ is properly defined and mathematically consistent.

(ii) We have that

$$m_0^{{\left({\eta }^{⊞\frac{1}{\gamma }}\right)}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }}=m_0^{\eta }=m_0^{{\left({\tilde{U}}^{\mathbf{t}}\right)}^{-1}\left({\varphi }^{-1}\left(D_{1/b}\left({\eta }^{⊞b}\right)\right)\right)}.$$

Then, $\epsilon >0$ exists so that ${\mathcal{V}}_{{\left({\eta }^{⊞\frac{1}{\gamma }}\right)}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }}(·)$ and ${\mathcal{V}}_{{\left({\tilde{U}}^{\mathbf{t}}\right)}^{-1}\left({\varphi }^{-1}\left(D_{1/b}\left({\eta }^{⊞b}\right)\right)\right)}(·)$ are defined on $(m_0^{\eta },m_0^{\eta }+\epsilon )$. Using (20) and (4), for $u\in (m_0^{\eta },m_0^{\eta }+\epsilon )$, we have

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{{\left({\eta }^{⊞\frac{1}{\gamma }}\right)}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }}\left(u\right)& =& \gamma {\mathcal{V}}_{{\eta }^{⊞\frac{1}{\gamma }}}(u/\gamma )+(1/\gamma -1)\left(u-\gamma m_0^{{\eta }^{⊞\frac{1}{\gamma }}}\right)\left(u(1-b)+\gamma (b-a)m_0^{{\eta }^{⊞\frac{1}{\gamma }}}\right)\hfill \\ & =& {\mathcal{V}}_{\eta }\left(u\right)+(1/\gamma -1)(u-m_0^{\eta })(u(1-b)+(b-a)m_0^{\eta })\hfill \\ & \stackrel{\gamma \to +\infty }{\to }& {\mathcal{V}}_{\eta }\left(u\right)-(u-m_0^{\eta })(u(1-b)+(b-a)m_0^{\eta }).\hfill \end{array}$$

(30)

On the other hand, with ${\varphi }^{-1}\left(x\right)=bx-(b-a)m_0^{\eta }$ and ${\left({\tilde{U}}^{\mathbf{t}}\right)}^{-1}$ the inverse of ${\tilde{U}}^{\mathbf{t}}$, we have

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{{\left({\tilde{U}}^{\mathbf{t}}\right)}^{-1}\left({\varphi }^{-1}\left(D_{1/b}\left({\eta }^{⊞b}\right)\right)\right)}\left(u\right)& =& {\displaystyle \frac{1}{b}}{\mathcal{V}}_{{\varphi }^{-1}\left(D_{1/b}\left({\eta }^{⊞b}\right)\right)}(bu+(a-b)m_0^{\eta })+(u-m_0^{\eta })(bu+(a-b)m_0^{\eta }-u)\hfill \\ & =& b{\mathcal{V}}_{D_{1/b}\left({\eta }^{⊞b}\right)}\left(u\right)+(u-m_0^{\eta })(bu+(a-b)m_0^{\eta }-u)\hfill \\ & =& {\mathcal{V}}_{\eta }\left(u\right)-(u-m_0^{\eta })(u(1-b)+(b-a)m_0^{\eta }).\hfill \end{array}$$

(31)

Relation (30) and (31) gives

$${\mathcal{V}}_{{\left({\eta }^{⊞\frac{1}{\gamma }}\right)}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }}\left(u\right)\stackrel{\gamma \to +\infty }{\to }{\mathcal{V}}_{{\left({\tilde{U}}^{\mathbf{t}}\right)}^{-1}\left({\varphi }^{-1}\left(D_{1/b}\left({\eta }^{⊞b}\right)\right)\right)}\left(u\right).$$

The demonstration of (25) is completed by [16] (Proposition 4.2).

Note that the condition $b\in [1,+\infty )$ ensure that measure ${\left({\tilde{U}}^{\mathbf{t}}\right)}^{-1}\left({\varphi }^{-1}\left(D_{1/b}\left({\eta }^{⊞b}\right)\right)\right)$ is well defined.

(iii) We have that

$$m_0^{{\left({\eta }^{{⊞}_{\left(\mathbf{t}\right)}\frac{1}{\gamma }}\right)}^{\uplus \gamma }}=m_0^{\eta }=m_0^{{\mathbb{B}}_1^{-1}\left(D_b{\left(\varphi \left({\tilde{U}}^{\mathbf{t}}\left(\eta \right)\right)\right)}^{⊞\frac{1}{b}}\right)}.$$

Then, $\epsilon >0$ exists so that ${\mathcal{V}}_{{\left({\eta }^{{⊞}_{\left(\mathbf{t}\right)}\frac{1}{\gamma }}\right)}^{\uplus \gamma }}(·)$ and ${\mathcal{V}}_{{\mathbb{B}}_1^{-1}\left(D_b{\left(\varphi \left({\tilde{U}}^{\mathbf{t}}\left(\eta \right)\right)\right)}^{⊞\frac{1}{b}}\right)}(·)$ are defined on $(m_0^{\eta },m_0^{\eta }+\epsilon )$.

Using (21), (23) and (4), for $u\in (m_0^{\eta },m_0^{\eta }+\epsilon )$, we have

$$\begin{array}{ll}{\mathcal{V}}_{{\left({\eta }^{{⊞}_{\left(\mathbf{t}\right)}\frac{1}{\gamma }}\right)}^{\uplus \gamma }}\left(u\right)=\hfill & \gamma {\mathcal{V}}_{{\eta }^{{⊞}_{\left(\mathbf{t}\right)}\frac{1}{\gamma }}}(u/\gamma )+u(u-\gamma m_0^{{\eta }^{{⊞}_{\left(\mathbf{t}\right)}\frac{1}{\gamma }}})(1/\gamma -1)\\ & =\gamma \left[\frac{{\mathcal{V}}_{\eta }\left(u\right)}{\gamma }+(\gamma -1)(\frac{u}{\gamma }-\frac{m_0^{\eta }}{\gamma })(\frac{u(1-b)}{\gamma }+\frac{(b-a)m_0^{\eta }}{\gamma })\right]+u(u-m_0^{\eta })(\frac{1}{\gamma }-1)\\ & ={\mathcal{V}}_{\eta }\left(u\right)+(1-1/\gamma )(u-m_0^{\eta })(u(1-b)+(b-a)m_0^{\eta })+u(u-m_0^{\eta })(1/\gamma -1)\\ & \hspace{1em}\stackrel{\gamma \to +\infty }{\to }{\mathcal{V}}_{\eta }\left(u\right)+(u-m_0^{\eta })(u(1-b)+(b-a)m_0^{\eta })-u(u-m_0^{\eta })={\mathcal{V}}_{{\mathbb{B}}_1^{-1}\left(D_b{\left(\varphi \left({\tilde{U}}^{\mathbf{t}}\left(\eta \right)\right)\right)}^{⊞\frac{1}{b}}\right)}\left(u\right).\end{array}$$

(32)

The demonstration of (26) is achieved by [16] (Proposition 4.2).

(iv) We have that

$$m_0^{{\left({\eta }^{\uplus \frac{1}{\gamma }}\right)}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }}=m_0^{\eta }=m_0^{{\mathbb{B}}_1\left({\tilde{U}}^{{\mathbf{t}}^{-1}}\left({\varphi }^{-1}\left(D_{1/b}\left({\eta }^{⊞b}\right)\right)\right)\right)}.$$

Then, $\epsilon >0$ exists so that ${\mathcal{V}}_{{\left({\eta }^{\uplus \frac{1}{\gamma }}\right)}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }}(·)$ and ${\mathcal{V}}_{{\mathbb{B}}_1\left({\tilde{U}}^{{\mathbf{t}}^{-1}}\left({\varphi }^{-1}\left(D_{1/b}\left({\eta }^{⊞b}\right)\right)\right)\right)}(·)$ are defined on $(m_0^{\eta },m_0^{\eta }+\epsilon )$. Using (21), (22) and (4), for $u\in (m_0^{\rho },m_0^{\rho }+\epsilon )$, we have

$$\begin{array}{ll}{\mathcal{V}}_{{\left({\eta }^{\uplus \frac{1}{\gamma }}\right)}^{{⊞}_{\left(\mathbf{t}\right)}\gamma }}\left(u\right)=\hfill & \gamma {\mathcal{V}}_{{\eta }^{\uplus \frac{1}{\gamma }}}(u/\gamma )+(1/\gamma -1)\left(u-\gamma m_0^{{\eta }^{⊞\frac{1}{\gamma }}}\right)\left(u(1-b)+\gamma (b-a)m_0^{{\eta }^{⊞\frac{1}{\gamma }}}\right)\\ & ={\mathcal{V}}_{\rho }\left(u\right)+u(u-m_0^{\eta })(1-1/\gamma )+(1/\gamma -1)(u-m_0^{\eta })(u(1-b)+(b-a)m_0^{\eta })\\ & \stackrel{\gamma \to +\infty }{\to }{\mathcal{V}}_{\eta }\left(u\right)+(u-m_0^{\eta })(u(1-b)+(b-a)m_0^{\eta })+u(u-m_0^{\eta })={\mathcal{V}}_{{\mathbb{B}}_1\left({\tilde{U}}^{{\mathbf{t}}^{-1}}\left({\varphi }^{-1}\left(D_{1/b}\left({\eta }^{⊞b}\right)\right)\right)\right)}\left(u\right).\end{array}$$

(33)

The demonstration of (27) is achieved by [16] (Proposition 4.2).

□

Next, we emphasize the significance of Theorem 3 by examining its implications for several concrete examples of probability measures. This approach not only illustrates the practical relevance of the theorem but also clarifies how the theoretical results manifest in specific and well-known distributions. By applying Theorem 3 to particular cases, we gain deeper insight into the structural behavior of the transformation and its impact on key measures within free probability.

Example 1.

Consider $sb=\frac{1}{2}{\delta }_{-1}+\frac{1}{2}{\delta }_1$.

(i) 

When $b\in (0,1]$, we have

$${\left(s{b}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }\stackrel{\gamma \to +\infty }{\to }f{b}_{(0,-b)}in distribution,$$

(34)

where $f{b}_{(\alpha ,\beta )}$ is the $\mathcal{FB}$ measure (6) with parameters $\alpha \in \mathbb{R}$ and $-1\le \beta <0$.

In fact, for u close to $m_0^{{\left(s{b}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }}=m_0^{sb}=0=m_0^{D_b{\left(\varphi \left({\tilde{U}}^{\mathit{t}}\left(sb\right)\right)\right)}^{⊞\frac{1}{b}}}$, one has from (29)

$${\mathcal{V}}_{{\left(s{b}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }}\left(m\right)\stackrel{\gamma \to +\infty }{\to }{\mathcal{V}}_{D_b{\left(\varphi \left({\tilde{U}}^{\mathit{t}}\left(sb\right)\right)\right)}^{⊞\frac{1}{b}}}\left(u\right)=1-b{u}^2={\mathcal{V}}_{f{b}_{(0,-b)}}\left(u\right).$$

(ii) 

When $b\in [1,+\infty )$, for u close to $m_0^{{\left(s{b}^{⊞\frac{1}{\gamma }}\right)}^{{⊞}_{\left(\mathit{t}\right)}\gamma }}=m_0^{sb}=0=m_0^{{\left({\tilde{U}}^{\mathit{t}}\right)}^{-1}\left({\varphi }^{-1}\left(D_{1/b}\left(s{b}^{⊞b}\right)\right)\right)}$, one has from (31)

$${\mathcal{V}}_{{\left(s{b}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }}\left(u\right)\stackrel{\gamma \to +\infty }{\to }{\mathcal{V}}_{{\left({\tilde{U}}^{\mathit{t}}\right)}^{-1}\left({\varphi }^{-1}\left(D_{1/b}\left(s{b}^{⊞b}\right)\right)\right)}\left(u\right)=1+(b-2)u^2.$$

• If $b\in [1,2)$, then $1+(b-2)u^2={\mathcal{V}}_{f{b}_{(0,b-2)}}\left(u\right)$ and in this case

  $${\left(s{b}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }\stackrel{\gamma \to +\infty }{\to }f{b}_{(0,b-2)}in distribution,$$
• If $b=2$, then $1+(b-2)u^2=1={\mathcal{V}}_{sc}\left(u\right)$ and in this case

  $${\left(s{b}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }\stackrel{\gamma \to +\infty }{\to }scin distribution,$$
  where $sc$ is the $\mathcal{SC}$ measure (9).
• If $b\in (2,+\infty )$, then

  $${\left(s{b}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }\stackrel{\gamma \to +\infty }{\to }f{h}_{(0,b-2)}in distribution,$$
  where $f{h}_{(\alpha ,\beta )}$ is the $\mathcal{FH}$ measure (6) with parameters $\beta >0$ and ${\alpha }^2<4\beta$.

Example 2.

Consider the $\mathcal{SC}$ measure (9).

(i) 

For $b\in (0,1]$, and u close to $m_0^{{\left(s{c}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }}=m_0^{sc}=0=m_0^{D_b{\left(\varphi \left({\tilde{U}}^{\mathit{t}}\left(sc\right)\right)\right)}^{⊞\frac{1}{b}}}$, one has from (31)

$${\mathcal{V}}_{{\left(s{c}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }}\left(u\right)\stackrel{\gamma \to +\infty }{\to }{\mathcal{V}}_{{\left({\tilde{U}}^{\mathit{t}}\right)}^{-1}\left({\varphi }^{-1}\left(D_{1/b}\left(s{c}^{⊞b}\right)\right)\right)}\left(u\right)=1+(1-b)u^2.$$

If $b=1$, then

$${\left(s{c}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }\stackrel{\gamma \to +\infty }{\to }scin distribution,$$

If $b\in (1,+\infty )$, then

$${\left(s{c}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }\stackrel{\gamma \to +\infty }{\to }f{h}_{(0,1-b)}in distribution.$$

(ii) 

For $b\in [1,+\infty )$, and u close to $m_0^{{\left(s{c}^{⊞\frac{1}{\gamma }}\right)}^{{⊞}_{\left(\mathit{t}\right)}\gamma }}=m_0^{sc}=0=m_0^{{\left({\tilde{U}}^{\mathit{t}}\right)}^{-1}\left({\varphi }^{-1}\left(D_{1/b}\left(s{c}^{⊞b}\right)\right)\right)}$, one has from (31)

$${\mathcal{V}}_{{\left(s{c}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }}\left(u\right)\stackrel{\gamma \to +\infty }{\to }{\mathcal{V}}_{{\left({\tilde{U}}^{\mathit{t}}\right)}^{-1}\left({\varphi }^{-1}\left(D_{1/b}\left(s{c}^{⊞b}\right)\right)\right)}\left(u\right)=1+(b-1)u^2.$$

If $b=1$, then

$${\left(s{c}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }\stackrel{\gamma \to +\infty }{\to }scin distribution,$$

If $b\in (1,+\infty )$, then

$${\left(s{c}^{{⊞}_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }\stackrel{\gamma \to +\infty }{\to }f{h}_{(0,b-1)}in distribution,$$

(iii) 

For $b\in (0,1]$, and u close to $m_0^{{\left(s{c}^{{\uplus }_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }}=m_0^{sc}=0=m_0^{{\mathbb{B}}_1^{-1}\left(D_b{\left(\varphi \left({\tilde{U}}^{\mathit{t}}\left(sc\right)\right)\right)}^{⊞\frac{1}{b}}\right)}$, one has

$${\mathcal{V}}_{{\left(s{c}^{{\uplus }_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }}\left(u\right)\stackrel{\gamma \to +\infty }{\to }{\mathcal{V}}_{{\mathbb{B}}_1^{-1}\left(D_b{\left(\varphi \left({\tilde{U}}^{\mathit{t}}\left(sc\right)\right)\right)}^{⊞\frac{1}{b}}\right)}\left(u\right)=1-b{u}^2.$$

Then

$${\left(s{c}^{{\uplus }_{\left(\mathit{t}\right)}\frac{1}{\gamma }}\right)}^{⊞\gamma }\stackrel{\gamma \to +\infty }{\to }f{b}_{(0,-b)}in distribution.$$

The study of the ${\mathbb{B}}_r$-transformation [21] and the $(a,b)$-transformation reflects two distinct but complementary approaches to deformation within non-commutative probability. The ${\mathbb{B}}_t$-transformation primarily acts as a Boolean-to-free interpolation mechanism, describing how a distribution evolves between Boolean and free convolution regimes while preserving the analytic structure of the CST. Its analysis within CSK families emphasizes invariance and stability properties, particularly how the kernel families associated with free Meixner and free Letac–Mora classes remain closed under such transformations. In contrast, the $(a,b)$-transformation introduces a two-parameter deformation directly on the generating measure of the CSK family, modifying both the scaling and the shift of the reciprocal CST. This leads to a richer geometric interpretation, capturing variations in mean and VFs and linking the analytic deformations to statistical parameters within the kernel family. Thus, while the ${\mathbb{B}}_t$-transformation captures the dynamic evolution of measures along a probabilistic interpolation axis, the $(a,b)$-transformation encodes structural and parametric deformations intrinsic to the CSK framework, offering a broader geometric and statistical interpretation.

5. Conclusions

In this article, we have presented a comprehensive analysis of the invariance of the FMF under $(a,b)$-deformations. The study offers a detailed clarification of how this invariance arises and emphasizes its implications for the structural stability and internal coherence of the FMF. By demonstrating that the fundamental characteristics of the family, such as its moments, cumulants, and VFs, remain preserved under deformation, we highlight the robustness of the FMF within the broader framework of free probability theory. Beyond the study of invariance, we also establish several new limiting theorems involving both Boolean and free additive convolutions. These results are formulated through the lens of the VF, which plays a central role in capturing the analytical relationships between deformations and convolutions. The use of the VF as an analytical bridge allows for a unified interpretation of stability and transformation behaviors across different probabilistic operations. By integrating these distinct perspectives, the present work provides a deeper and more comprehensive understanding of how $(a,b)$-deformations interact with fundamental operations in free probability. The results not only generalize and refine existing theoretical findings but also uncover new connections between algebraic symmetries and analytical properties of the FMF.

While the $(a,b)$-transformation provides a powerful analytical tool for exploring deformations within CSK families, certain methodological limitations remain. First, most current analyses rely on specific parametric or regularity assumptions on the generating measures, which may restrict the applicability of the transformation to a limited class of CSK families. Moreover, the interaction between the two parameters a and b, particularly in non-linear regimes, can lead to intricate analytic behavior of the CST, making it difficult to obtain explicit expressions or asymptotic descriptions. Another limitation lies in the lack of a complete probabilistic interpretation of the $(a,b)$-deformation in terms of convolution operations or stochastic models, unlike the ${\mathbb{B}}_r$-transformation, whose Boolean-free interpolation has a clear operational meaning.

Future research should aim to extend the theoretical framework of the $(a,b)$-transformation to encompass broader classes of measures, possibly relaxing smoothness or compactness conditions. Investigating asymptotic behaviors and fixed points of the transformation could also shed light on new universality phenomena within CSK families. Moreover, establishing connections with information geometry and statistical inference may provide a deeper understanding of the geometric role of parameters a and b as deformations of mean and VFs. Finally, developing computational and numerical techniques for approximating the deformed kernels would facilitate the application of the $(a,b)$-transformation to practical problems in random matrix theory and non-commutative statistics.