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Symmetry
Volume 17
Issue 11
10.3390/sym17111954
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Article

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

by
Abdulmajeed Albarrak
1,
Raouf Fakhfakh
1,* and
Ghadah Alomani
2
1
Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72388, Saudi Arabia
2
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
*
Author to whom correspondence should be addressed.
Symmetry 2025, 17(11), 1954; https://doi.org/10.3390/sym17111954 (registering DOI)
Submission received: 14 October 2025 / Revised: 8 November 2025 / Accepted: 10 November 2025 / Published: 13 November 2025
(This article belongs to the Section Mathematics)
Versions Notes

Abstract

This study is devoted to the detailed examination of the concept of (a,b)-deformation, defined for parameters a∈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.
Keywords: Cauchy transform; free Meixner family; (a,b)-deformation; variance function Cauchy transform; free Meixner family; (a,b)-deformation; variance function

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MDPI and ACS Style

Albarrak, A.; Fakhfakh, R.; Alomani, G. A Study of the (a,b)-Deformed Free Convolution. Symmetry 2025, 17, 1954. https://doi.org/10.3390/sym17111954

AMA Style

Albarrak A, Fakhfakh R, Alomani G. A Study of the (a,b)-Deformed Free Convolution. Symmetry. 2025; 17(11):1954. https://doi.org/10.3390/sym17111954

Chicago/Turabian Style

Albarrak, Abdulmajeed, Raouf Fakhfakh, and Ghadah Alomani. 2025. "A Study of the (a,b)-Deformed Free Convolution" Symmetry 17, no. 11: 1954. https://doi.org/10.3390/sym17111954

APA Style

Albarrak, A., Fakhfakh, R., & Alomani, G. (2025). A Study of the (a,b)-Deformed Free Convolution. Symmetry, 17(11), 1954. https://doi.org/10.3390/sym17111954

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