On Generalized <i>t</i>-Transformation of Free Convolution
A notion of generalized (two-parameterized) <i>t</i>-transformation of free convolution, also called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant=&qu...
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| Language: | English |
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MDPI AG
2024-03-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/16/3/372 |
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| author | Ayed. R. A. Alanzi Raouf Fakhfakh Fatimah Alshahrani |
| author_facet | Ayed. R. A. Alanzi Raouf Fakhfakh Fatimah Alshahrani |
| author_sort | Ayed. R. A. Alanzi |
| collection | DOAJ |
| description | A notion of generalized (two-parameterized) <i>t</i>-transformation of free convolution, also called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="bold">t</mi><mo>=</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>-deformed free convolution, is introduced for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. In this article, some results of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">t</mi></semantics></math></inline-formula>-deformed free convolution are given within the theory of Cauchy-Stieltjes Kernel (CSK) families. The variance function is a fundamental concept in CSK families. An expression is provided for the variance function under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">t</mi></semantics></math></inline-formula>-deformed free convolution power. In addition, through the use of the variance function, an approximation is provided for members of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">t</mi></semantics></math></inline-formula>-deformed free Gaussian CSK family and members of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">t</mi></semantics></math></inline-formula>-deformed free Poisson CSK family respectively. Furthermore, by involving the free multiplicative convolution, a new limit theorem is provided with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">t</mi></semantics></math></inline-formula>-deformed free convolution. |
| first_indexed | 2024-04-24T17:48:33Z |
| format | Article |
| id | doaj.art-99bfe79e253d4df3a91d557492fc2fbc |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-24T17:48:33Z |
| publishDate | 2024-03-01 |
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| series | Symmetry |
| spelling | doaj.art-99bfe79e253d4df3a91d557492fc2fbc2024-03-27T14:05:37ZengMDPI AGSymmetry2073-89942024-03-0116337210.3390/sym16030372On Generalized <i>t</i>-Transformation of Free ConvolutionAyed. R. A. Alanzi0Raouf Fakhfakh1Fatimah Alshahrani2Department of Mathematics, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi ArabiaDepartment of Mathematics, College of Science and Arts in Gurayat, Jouf University, Gurayat 77454, Saudi ArabiaDepartment of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi ArabiaA notion of generalized (two-parameterized) <i>t</i>-transformation of free convolution, also called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="bold">t</mi><mo>=</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>-deformed free convolution, is introduced for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. In this article, some results of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">t</mi></semantics></math></inline-formula>-deformed free convolution are given within the theory of Cauchy-Stieltjes Kernel (CSK) families. The variance function is a fundamental concept in CSK families. An expression is provided for the variance function under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">t</mi></semantics></math></inline-formula>-deformed free convolution power. In addition, through the use of the variance function, an approximation is provided for members of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">t</mi></semantics></math></inline-formula>-deformed free Gaussian CSK family and members of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">t</mi></semantics></math></inline-formula>-deformed free Poisson CSK family respectively. Furthermore, by involving the free multiplicative convolution, a new limit theorem is provided with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">t</mi></semantics></math></inline-formula>-deformed free convolution.https://www.mdpi.com/2073-8994/16/3/372variance functionCauchy–Stieltjes transformdeformation of measures |
| spellingShingle | Ayed. R. A. Alanzi Raouf Fakhfakh Fatimah Alshahrani On Generalized <i>t</i>-Transformation of Free Convolution Symmetry variance function Cauchy–Stieltjes transform deformation of measures |
| title | On Generalized <i>t</i>-Transformation of Free Convolution |
| title_full | On Generalized <i>t</i>-Transformation of Free Convolution |
| title_fullStr | On Generalized <i>t</i>-Transformation of Free Convolution |
| title_full_unstemmed | On Generalized <i>t</i>-Transformation of Free Convolution |
| title_short | On Generalized <i>t</i>-Transformation of Free Convolution |
| title_sort | on generalized i t i transformation of free convolution |
| topic | variance function Cauchy–Stieltjes transform deformation of measures |
| url | https://www.mdpi.com/2073-8994/16/3/372 |
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