Erdélyi-Kober fractional integral operators from a statistical perspective (I)

Abstract

In this article we examine the densities of a product and a ratio of two real positive scalar random variables x1 and x2, which are statistically independently distributed, and we consider the density of the product u1 = x1x2 as well as the density of the ratio u2 = and show that Kober operator of the second kind is available as the density of u1 and Kober operator of the first kind is available as the density of u2 when x1 has a type-1 beta density and x2 has an arbitrary density. We also give interpretations of Kober operators of the second and first kind as Mellin convolution for a product and ratio respectively. Then we look at various types of generalizations of the idea thereby obtaining a large collection of operators which can all be called generalized Kober operators. One of the generalizations considered is the pathway idea where one can move from one family of operators to another family and yet another family and eventually end up with an exponential form. Common generalizations in terms of a Gauss' hypergeometric series is also given a statistical interpretation and put on a more general structure so that the standard generalizations given by various authors, including Saigo operators, are given statistical interpretations and are derivable as special cases of the general structure considered in this article.

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

“…This memory kernel is determined by the specific distribution of the diffusion coefficients f ( λ ). Moreover, because of this memory kernel, it follows that the resulting evolution equation is a fractional differential equation, and a special connection exists between the generalized Gamma distribution and the Erdélyi–Kober fractional operators (see [42,53,54]).…”

Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models

“…In application of the generalized and general fractional operators, an important question arises about the correct subject interpretation of these operators (for example, see informational [38], physical [39], and economic [40][41][42] interpretations). It is important to emphasize that not all fractional operators can describe the processes with memory (for example, see [43][44][45][46]). It is important to clearly understand what type of phenomena a given operator can describe.…”

“…The Kober fractional integration of non-integer order [1,2,4] can be interpreted as an expected value of a random variable up to a constant factor (for example, see [43,45] and section 10 in [46]), where the random variable describes scaling (dilation) with the gamma distribution. The Erdelyi-Kober integral operator, the differential operators of Kober and Erdelyi-Kober type have analogous interpretation [43,45,46]. As a result, these operators are integer-order operator with continuously distributed scaling (dilation), and these operators cannot describe the memory.…”

“…We see that the Kober integral operator describes beta distributed scaling up to numerical factor. For details see [43,45] and section 10 in [46].…”

Stochastic Processes via the Pathway Model

“…Thus, essentially, all fractional integral operators belong to the categories of Mellin convolution of a D n f ] in the Caputo sense (25) for i = 1, 2, see also [36,37].…”