The characteristic (Laplace or Levy) exponents uniquely characterize infinitely divisible probability distributions. Although of purely mathematical origin they appear to be uniquely associated with the memory functions present in evolution equations
which govern the course of such physical phenomena like non-Debye relaxations or anomalous diffusion. Commonly accepted procedure to mimic memory effects is to make basic equations time smeared, i.e., nonlocal in time. This is modeled either through the convolution of memory functions with those describing relaxation/diffusion or, alternatively, through the time smearing of time derivatives. Intuitive expectations say that such introduced time smearings should be physically equivalent. This leads to the conclusion that both kinds of so far introduced memory functions form a twin structure familiar to mathematicians for a long time and known as the Sonine pair. As an illustration of the proposed scheme we consider the excess wings model of non-Debye relaxations, determine its evolution equations and discuss properties of the solutions.
The various types of generalized Cattaneo, called also telegraphers equation, are studied. We find conditions under which solutions of the equations considered so far can be recognized as probability distributions, textit{i.e.} are normalizable and n
on-negative on their domains. Analysis of the relevant mean squared displacements enables us to classify diffusion processes described by such obtained solutions and to identify them with either ordinary or anomalous super- or subdiffusion. To complete our study we analyse derivations of just considered examples the generalized Cattaneo equations using the continuous time random walk and the persistent random walk approaches.
We calculate exactly the Laplace transform of the Fr{e}chet distribution in the form $gamma x^{-(1+gamma)} exp(-x^{-gamma})$, $gamma > 0$, $0 leq x < infty$, for arbitrary rational values of the shape parameter $gamma$, i.e. for $gamma = l/k$ with $l
, k = 1,2, ldots$. The method employs the inverse Mellin transform. The closed form expressions are obtained in terms of Meijer G functions and their graphical illustrations are provided. A rescaled Fr{e}chet distribution serves as a kernel of Fr{e}chet integral transform. It turns out that the Fr{e}chet transform of one-sided L{e}vy law reproduces the Fr{e}chet distribution.
Recent experiments have shown that photoluminescence decay of silicon nanocrystals can be described by the stretched exponential function. We show here that the associated decay probability rate is the one-sided Levy stable distribution which describ
es well the experimental data. The relevance of these conclusions to the underlying stochastic processes is discussed in terms of Levy processes.
Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec [J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of coherent states, related to the Hermite polynomials in a complex variable which are orthogonal
with respect to a non-rotationally invariant measure. We investigate relations between these coherent states and obtain the relationship between them and the squeezed states of quantum optics. We also obtain a second realization of the canonical coherent states in the Bargmann space of analytic functions, in terms of a squeezed basis. All this is done in the flavor of the classical approach of V. Bargmann [Commun. Pur. Appl. Math. 14, 187 (1961)].
We study the higher-order heat-type equation with first time and M-th spatial partial derivatives, M = 2, 3, ... . We demonstrate that its exact solutions for M even can be constructed with the help of signed Levy stable functions. For M odd the same
role is played by a special generalization of Airy Ai function that we introduce and study. This permits one to generate the exact and explicit heat kernels pertaining to these equations. We examine analytically and graphically the spacial and temporary evolution of particular solutions for simple initial conditions.
We study integral representation of so-called $d$-dimensional Catalan numbers $C_{d}(n)$, defined by $[prod_{p=0}^{d-1} frac{p!}{(n+p)!}] (d n)!$, $d = 2, 3, ...$, $n=0, 1, ...$. We prove that the $C_{d}(n)$s are the $n$th Hausdorff power moments of
positive functions $W_{d}(x)$ defined on $xin[0, d^d]$. We construct exact and explicit forms of $W_{d}(x)$ and demonstrate that they can be expressed as combinations of $d-1$ hypergeometric functions of type $_{d-1}F_{d-2}$ of argument $x/d^d$. These solutions are unique. We analyse them analytically and graphically. A combinatorially relevant, specific extension of $C_{d}(n)$ for $d$ even in the form $D_{d}(n)=[prod_{p = 0}^{d-1} frac{p!}{(n+p)!}] [prod_{q = 0}^{d/2 - 1} frac{(2 n + 2 q)!}{(2 q)!}]$ is analyzed along the same lines.
Symbolic methods of umbral nature play an important and increasing role in the theory of special functions and in related fields like combinatorics. We discuss an application of these methods to the theory of lacunary generating functions for the Lag
uerre polynomials for which we give a number of new closed form expressions. We present furthermore the different possibilities offered by the method we have developed, with particular emphasis on their link to a new family of special functions and with previous formulations, associated with the theory of quasi monomials.
We review some aspects of the theory of spherical Bessel functions and Struve functions by means of an operational procedure essentially of umbral nature, capable of providing the straightforward evaluation of their definite integrals and of successi
ve derivatives. The method we propose allows indeed the formal reduction of these family of functions to elementary ones of Gaussian type. We study the problem in general terms and present a formalism capable of providing a unifying point of view including Anger and Weber functions too. The link to the multi-index Bessel functions is also briefly discussed.
We present exact and explicit form of the kernels $hat{K}(x, y)$ appearing in the theory of energy correlations in the ensembles of Hermitian random matrices with Gaussian probability distribution, see E. Brezin and S. Hikami, Phys. Rev. E 57, 4140 a
nd E 58, 7176 (1998). In obtaining this result we have exploited the analogy with the method of producing exact forms of two-sided, symmetric Levy stable laws, presented by us recently. This result is valid for arbitrary values of parameters in question. We furnish analytical and graphical representations of physical quantities calculated from $hat{K}(x, y)$s.
