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.
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.
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.
We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Levy stable probability distributions g_{alpha}(x), 0 leq x < infty, 0 < alpha < 1. We demonstrate that the knowledge of one such a distribution g_{alpha}(x) suf
fices to obtain exactly g_{alpha^{p}}(x), p=2, 3,... Similarly, from known g_{alpha}(x) and g_{beta}(x), 0 < alpha, beta < 1, we obtain g_{alpha beta}(x). The method is based on the construction of the integral operator, called Levy transform, which implements the above operations. For alpha rational, alpha = l/k with l < k, we reproduce in this manner many of the recently obtained exact results for g_{l/k}(x). This approach can be also recast as an application of the Efros theorem for generalized Laplace convolutions. It relies solely on efficient definite integration.
Generalized Bargmann representations which are based on generalized coherent states are considered. The growth of the corresponding analytic functions in the complex plane is studied. Results about the overcompleteness or undercompleteness of discret
e sets of these generalized coherent states are given. Several examples are discussed in detail.
We obtain exact results for fractional equations of Fokker-Planck type using evolution operator method. We employ exact forms of one-sided Levy stable distributions to generate a set of self-reproducing solutions. Explicit cases are reported and stud
ied for various fractional order of derivatives, different initial conditions, and for differe
We study the one-dimensional Levy stable density distributions g(alpha, beta; x) for -infty < x < infty, for rational values of index alpha and the asymmetry parameter beta: alpha = l/k and beta = (l - 2r)/k, where l, k and r are positive integers su
ch that 0 < l/k < 1 for 0 <= r <= l and 1 < l/k <= 2 for l-k <= r <= k. We treat both symmetric (beta = 0) and asymmetric (beta neq 0) cases. We furnish exact and explicit forms of g(alpha, beta; x) in terms of known functions for any admissible values of alpha and beta specified by a triple of integers k, l and r. We reproduce all the previously known exact results and we study analytically and graphically many new examples. We point out instances of experimental and statistical data that could be described by our solutions.
We consider discrete spectra of bound states for non-relativistic motion in attractive potentials V_{sigma}(x) = -|V_{0}| |x|^{-sigma}, 0 < sigma leq 2. For these potentials the quasiclassical approximation for n -> infty predicts quantized energy le
vels e_{sigma}(n) of a bounded spectrum varying as e_{sigma}(n) ~ -n^{-2sigma/(2-sigma)}. We construct collective quantum states using the set of wavefunctions of the discrete spectrum taking into account this asymptotic behaviour. We give examples of states that are normalizable and satisfy the resolution of unity, using explicit positive functions. These are coherent states in the sense of Klauder and their completeness is achieved via exact solutions of Hausdorff moment problems, obtained by combining Laplace and Mellin transform methods. For sigma in the range 0<sigmaleq 2/3 we present exact implementations of such states for the parametrization sigma = 2(k-l)/(3k-l), with k and l positive integers satisfying k>l.
We obtain and investigate the regular eigenfunctions of simple differential operators x^r d^{r+1}/dx^{r+1}, r=1, 2, ... with the eigenvalues equal to one. With the help of these eigenfunctions we construct a non-unitary analogue of boson displacement
operator which will be acting on the vacuum. In this way we generate collective quantum states of the Fock space which are normalized and equipped with the resolution of unity with the positive weight functions that we obtain explicitly. These states are thus coherent states in the sense of Klauder. They span the truncated Fock space without first r lowest-lying basis states: |0>, |1>, ..., |r-1>. These states are squeezed, are sub-Poissonian in nature and are reminiscent of photon-added states at Agarwal et al.
