Stability and properties of Cauchy–Stieltjes Kernel families under generalized $ t $-transformation

Fatimah Alshahrani; Raouf Fakhfakh; Fatimah Alshahrani; Raouf Fakhfakh

doi:10.3934/math.2025939

AIMS Mathematics

2025, Volume 10, Issue 9: 21025-21039. doi: 10.3934/math.2025939

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Research article

Stability and properties of Cauchy–Stieltjes Kernel families under generalized $ t $-transformation

Fatimah Alshahrani ^{1
,},
Raouf Fakhfakh ^{2
,
,}

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia; fmalshahrani@pnu.edu.sa
2.
Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72388, Saudi Arabia; rfakhfakh@ju.edu.sa

Received: 10 May 2025 Revised: 22 August 2025 Accepted: 01 September 2025 Published: 12 September 2025
MSC : 46L54, 60E10

The study of the generalized $ t $-deformation of free convolution from the lens of Cauchy–Stieltjes kernel (CSK) families provides an excellent mathematical framework for understanding noncommutative probability distributions. In this paper, we use the concept of generalized $ t $-deformation to demonstrate different elements of the Marchenko–Pastur, free Gamma, and inverse semicircle measures in the CSK families setting. These findings advance our knowledge of generalized $ t $-deformation in the non-commutative probability framework.
- Marchenko–Pastur distribution,
- free Gamma distribution,
- variance function,
- inverse semicircle distribution,
- Cauchy–Stieltjes transform
Citation: Fatimah Alshahrani, Raouf Fakhfakh. Stability and properties of Cauchy–Stieltjes Kernel families under generalized $ t $-transformation[J]. AIMS Mathematics, 2025, 10(9): 21025-21039. doi: 10.3934/math.2025939

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Abstract

The study of the generalized $ t $-deformation of free convolution from the lens of Cauchy–Stieltjes kernel (CSK) families provides an excellent mathematical framework for understanding noncommutative probability distributions. In this paper, we use the concept of generalized $ t $-deformation to demonstrate different elements of the Marchenko–Pastur, free Gamma, and inverse semicircle measures in the CSK families setting. These findings advance our knowledge of generalized $ t $-deformation in the non-commutative probability framework.

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