Generalization of the Alpha-Stable Distribution with the Degree of Freedom

A Wright function based framework is proposed to combine and extend several distribution families. The $\alpha$-stable distribution is generalized by adding the degree of freedom parameter. The PDF of this two-sided super distribution family subsumes those of the original $\alpha$-stable, Student's t distributions, as well as the exponential power distribution and the modified Bessel function of the second kind. Its CDF leads to a fractional extension of the Gauss hypergeometric function. The degree of freedom makes possible for valid variance, skewness, and kurtosis, just like Student's t. The original $\alpha$-stable distribution is viewed as having one degree of freedom, that explains why it lacks most of the moments. A skew-Gaussian kernel is derived from the characteristic function of the $\alpha$-stable law, which maximally preserves the law in the new framework. To facilitate such framework, the stable count distribution is generalized as the fractional extension of the generalized gamma distribution. It provides rich subordination capabilities, one of which is the fractional $\chi$ distribution that supplies the needed 'degree of freedom' parameter. Hence, the "new" $\alpha$-stable distribution is a "ratio distribution" of the skew-Gaussian kernel and the fractional $\chi$ distribution. Mathematically, it is a new form of higher transcendental function under the Wright function family. Last, the new univariate symmetric distribution is extended to the multivariate elliptical distribution successfully.