{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T01:22:55Z","timestamp":1773451375849,"version":"3.50.1"},"reference-count":17,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,19]],"date-time":"2024-03-19T00:00:00Z","timestamp":1710806400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","award":["PNURSP2024R358"],"award-info":[{"award-number":["PNURSP2024R358"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A notion of generalized (two-parameterized) t-transformation of free convolution, also called (t=(a,b))-deformed free convolution, is introduced for a\u2208R and b&gt;0. In this article, some results of t-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 t-deformed free convolution power. In addition, through the use of the variance function, an approximation is provided for members of the t-deformed free Gaussian CSK family and members of the t-deformed free Poisson CSK family respectively. Furthermore, by involving the free multiplicative convolution, a new limit theorem is provided with respect to t-deformed free convolution.<\/jats:p>","DOI":"10.3390\/sym16030372","type":"journal-article","created":{"date-parts":[[2024,3,19]],"date-time":"2024-03-19T09:39:31Z","timestamp":1710841171000},"page":"372","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["On Generalized t-Transformation of Free Convolution"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5337-2276","authenticated-orcid":false,"given":"Ayed. R. A.","family":"Alanzi","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia"},{"name":"Department of Mathematics, College of Science and Arts in Gurayat, Jouf University, Gurayat 77454, Saudi Arabia"}]},{"given":"Raouf","family":"Fakhfakh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Arts in Gurayat, Jouf University, Gurayat 77454, Saudi Arabia"},{"name":"Laboratory of Probability and Statistics, Faculty of Sciences of Sfax, University of Sfax, Sfax 3000, Tunisia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1306-7688","authenticated-orcid":false,"given":"Fatimah","family":"Alshahrani","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"95","DOI":"10.4064\/-43-1-95-103","article-title":"New examples of onvolutions and non-commutative central limit theorems","volume":"43","year":"1998","journal-title":"Banach Cent. Publ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1016\/S0246-0203(01)01084-6","article-title":"Remarks on t-transformations of measures and convolutions","volume":"37","year":"2001","journal-title":"Ann. Inst. Henri Poincar Probab. Statist."},{"key":"ref_3","first-page":"279","article-title":"Bargmann measures for t-deformed probability","volume":"34","author":"Krystek","year":"2014","journal-title":"Probab. Math. Stat."},{"key":"ref_4","first-page":"293","article-title":"Interpolations of Bargmann Type Measures","volume":"49","author":"Asai","year":"2016","journal-title":"Demonstr. 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[Ph.D. Thesis, Indiana University]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/3\/372\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:15:58Z","timestamp":1760105758000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/3\/372"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,19]]},"references-count":17,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["sym16030372"],"URL":"https:\/\/doi.org\/10.3390\/sym16030372","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,19]]}}}