| Summary: | 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.
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