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.

The Fokker–Planck equation of the superstatistical fractional Brownian motion with application to passive tracers inside cytoplasm

“…r ðuÞ: ð3:10Þ By using formula (3.9), the representation in terms of the H-Fox function [40,41] of the PDF (3.10) results to be…”

“…We observe that molecules’ PDF (3.1) can be rewritten as follows: Pfalse(x;tfalse)=ρΓ(ν/ρ)14πλ0t2H Zρν−1/2(x24λ0t2H),where Zρνfalse(⋅false) is the Krätzel function [40] Zρνfalse(ufalse)=∫0∞λν−1normale−false(u/λfalse)−λρ dλ,1emu>0,and the variance of particle displacement is expressed as follows: right left.5emthickmathspacetrue⟨x2⟩=2∫0∞x2P(x;t) normaldx...…”

“…Here, we found that the PDF of molecule displacement belongs to the family of the Krätzel special function [40], and its Fokker–Planck equation is a fractional diffusion equation in the Erdélyi–Kober sense [41,42] that cannot be reduced to existing models for anomalous diffusion.…”

Mixed Poisson process with Stacy mixing variable -- Working version

“…Let us note that in 2014 Princy [14] introduces Krätzel distribution and describes its relation with the Generalized Gamma distribution. In 2019 Kudryavtsev [22] defines Gamma-exponential distribution (GE distribution).…”

“…ρ ∈ (0, ∞), ν ∈ C, and when ρ ≤ 0, Re(ν) < 0, the Krätzel function investigated for example in Krätzel [12], Kilbas et al [13] and Princy [14]. The notation η ∈ Exp(1) means that the random variable (r.v.)…”