Volume 36, Issue 1 - Special Issue: Fractional Calculus and Fractional Differential Equations with Applications – Part 1. Dedicated to the memory of Prof. Anatoly Kilbas (1948–2010). Issue Editor: Sergei Rogosin

In this paper, we use the representation of the generalized hypergeometric function pFq$\,_pF_q$ in terms of the known Meijer G -function to extend the range of parameters of such a special function so it be convergent. Also, we establish the corresponding series representation for such an extension. In particular, we extend the hypergeometric functions 2F1$\,_2F_1$ and 3F2$\,_3F_2$ from |z|<1${|z|&lt;1}$ to |z|>1${|z|&gt;1}$. Moreover, we obtain asymptotic formulas to estimate the extended pFq$\,_pF_q$ at infinity.

It is shown that Mellin convolutions of products and ratios in the real scalar variable case can be considered as densities of products and ratios of two independently distributed real scalar positive random variables. It is also shown that these are also connected to Krätzel integrals and to the Krätzel transform in applied analysis, to reaction-rate probability integrals in astrophysics and to other related aspects when the random variables have gamma or generalized gamma densities, and to fractional calculus when one of the variables has a type-1 beta density and the other variable has an arbitrary density. Matrix-variate analogues are also discussed. In the matrix-variate case, the M-convolutions introduced by the author are shown to be directly connected to densities of products and ratios of statistically independently distributed positive definite matrix random variables in the real case and to Hermitian positive definite matrices in the complex domain. These M-convolutions reduce to Mellin convolutions in the scalar variable case.

We prove that the coefficients aij(x)${a_{ij}(x)}$, q ( x ) and the domain Ω of a multidimensional fractional diffusion equation can be recovered uniquely from measurements u(b,t)${u(b,t)}$, t∈(t0,t1)${t\in (t_0,t_1)}$, at an arbitrary single point b inside a bounded domain Ω⊂ℝn${\Omega \subset \mathbb {R}^n}$. From the measurements we first recover infinitely many spectral data (λm,ϕm(x))${(\lambda _m,\varphi _m(x))}$ of the elliptic operator associated with the fractional diffusion equation. Then, the coefficients aij(x)${a_{ij}(x)}$ and q ( x ) are found from linear algebraic systems of the form Ay=b${Ay = b}$, where A is a generalized Wronskian of some set of eigenfunctions that can be shown to be nontrivial. The domain Ω is reconstructed using the first eigenfunction ϕ1(x)${\varphi _1(x)}$.

Two differential transforms involving the Gauss hypergeometric function in the kernels are considered. They generalize the classical Riemann–Liouville and Erdélyi–Kober fractional differential operators. Formulas of compositions for such generalized fractional differentials with the product of Bessel functions of the first kind are proved. Special cases of products of cosine and sine functions are given. The results are established in terms of a generalized Lauricella function due to Srivastava and Daoust. Corresponding assertions for the Riemann–Liouville and the Erdélyi–Kober fractional integral transforms are presented. Statistical interpretations of fractional-order integrals and derivatives are also established.

The mathematical analysis of time flow in physical many-body systems leads to the study of long-time limits. This article discusses the interdisciplinary problem of local stationarity, how stationary solutions can remain slowly time dependent after a long-time limit. A mathematical definition of almost invariant and nearly indistinguishable states on C*-algebras is introduced using functions of bounded mean oscillation. Rescaling of time yields generalized time flows of almost invariant and macroscopically indistinguishable states, that are mathematically related to stable convolution semigroups and fractional calculus. The infinitesimal generator is a fractional derivative of order less than or equal to unity. Applications of the analysis are given to irreversibility and to a physical experiment.