Krätzel Function and Related Statistical Distributions

Abstract

The Krätzel function has many applications in applied analysis, so this function is used as a base to create a density function which will be called the Krätzel density. This density is applicable in chemical physics, Hartree–Fock energy, helium isoelectric series, statistical mechanics, nuclear energy generation, etc., and also connected to Bessel functions. The main properties of this new family are studied, showing in particular that it may be generated via mixtures of gamma random variables. Some basic statistical quantities associated with this density function such as moments, Mellin transform, and Laplace transform are obtained. Connection of Krätzel distribution to reaction rate probability integral in physics, inverse Gaussian density in stochastic processes, Tsallis statistics and superstatistics in non-extensive statistical mechanics, Mellin convolutions of products and ratios thereby to fractional integrals, synthetic aperture radar, and other areas are pointed out in this article. Finally, we extend the Krätzel density using the pathway model of Mathai, and some applications are also discussed. The new probability model is fitted to solar radiation data.

Keywords

Laplace transform / Krätzel function / Mellin transform / H-function / Krätzel density / Pathway model

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