For the first time, an explicit closed form expression is derived for the characteristic function of the generalized normal distribution (GND). Also derived is an expression for the correlation coefficient between variate-values and their ranks in samples from the GND. The expression for the former involves the Fox-Wright generalized confluent hypergeometric 1Ψ0-function, while the latter is ex...

We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.

A characterization theorem based on first moment is given for the subfamily of distributions generated by the Gaussian hypergeometric function. The theorem is then applied to some discrete probability distributions, providing specific characterization theorems for each of them.

We derive simple linear, inhomogeneous recurrences for the variance of the index by utilising the fact that the generating function for the distribution of the number of positive eigenvalues of a Gaussian unitary ensemble is a τ-function of the fourth Painlevé equation. From this we deduce a simple summation formula, several integral representations and finally an exact hypergeometric function ...

Sanford S. Miller and Petru T. Mocanu’s theory of second-order differential subordinations was extended for the case third-order by José A. Antonino in 2011. In this paper, new results are proved regarding that extend ones involving classical subordination theory. A method finding a dominant is provided when behavior function not known on boundary unit disc. Additionally, obtaining best present...

A concise method of deriving expressions for Gaussian-like solutions the paraxial and wave equation (i.e. 3D d’Alembert equation) is presented. This based on Hankel transform. Choosing some Gaussian base functions with slight modifications prefactor all basic beams cylindrical character can be easily obtained. refers to Gaussian, Bessel-Gaussian, modified Laguerre–Gaussian Kummer-Gaussian (i.e....

In the framework of the paraxial approximation, the analytical expression of a family of Hypergeometric-Gaussian beams type-II passing through a paraxial ABCD optical system with a circular annular aperture is derived by means the expansion of the hard aperture function into a finite sum of complex Gaussian functions and using the generalized Huygens-Fresnel diffraction integral. Based on the o...

Carlson and Shaffer [B.C. Carlson, D.B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984) 737–745] have introduced a linear operator associated with the Gaussian hypergeometric function which has been generalized by Dziok and Srivastava [J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl....