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K. Górska

Publications and source records attributed to K. Górska.

36 records · Page 2Linked to original sources
arXiv2019

Operational versus umbral methods and the Borel transform

Integro-differential methods, currently exploited in calculus, provide an inexhaustible source of tools to be applied to a wide class of problems, involving the theory of special functions and other subjects. The use of integral transforms of the Borel type and the associated formalism is shown to be a very effective mean, constituting a solid bridge between umbral and operational methods. We merge these different points of view to obtain new and efficient analytical techniques for the derivation of integrals of special functions and the summation of associated generating functions as well.

math.CA↗
arXiv2018

Composition law for the Cole-Cole relaxation and ensuing evolution equations

Physically natural assumption says that the any relaxation process taking place in the time interval $[t_{0},t_{2}]$, $t_{2} > t_{0}\ge 0$ may be represented as a composition of processes taking place during time intervals $[t_{0}, t_{1}]$ and $[t_{1},t_{2}]$ where $t_{1}$ is an arbitrary instant of time such that $t_{0} \leq t_{1} \leq t_{2}$. For the Debye relaxation such a composition is realized by usual multiplication which claim is not valid any longer for more advanced models of relaxation processes. We investigate the composition law required to be satisfied by the Cole-Cole relaxation and find its explicit form given by an integro-differential relation playing the role of the time evolution equation. The latter leads to differential equations involving fractional derivatives, either of the Caputo or the Riemann-Liouville senses, which are equivalent to the special case of the fractional Fokker-Planck equation satisfied by the Mittag-Leffler function known to describe the Cole-Cole relaxation in the time domain.

cond-mat.mes-hall↗
arXiv2018

Holomorphic Hermite polynomials in two variables

Generalizations of the Hermite polynomials to many variables and/or to the complex domain have been located in mathematical and physical literature for some decades. Polynomials traditionally called complex Hermite ones are mostly understood as polynomials in $z$ and $\bar{z}$ which in fact makes them polynomials in two real variables with complex coefficients. The present paper proposes to investigate for the first time holomorphic Hermite polynomials in two variables. Their algebraic and analytic properties are developed here. While the algebraic properties do not differ too much for those considered so far, their analytic features are based on a kind of non-rotational orthogonality invented by van Eijndhoven and Meyers. Inspired by their invention we merely follow the idea of Bargmann's seminal paper (1961) giving explicit construction of reproducing kernel Hilbert spaces based on those polynomials. "Homotopic" behavior of our new formation culminates in comparing it to the very classical Bargmann space of two variables on one edge and the aforementioned Hermite polynomials in $z$ and $\bar{z}$ on the other. Unlike in the case of Bargmann's basis our Hermite polynomials are not product ones but factorize to it when bonded together with the first case of limit properties leading both to the Bargmann basis and suitable form of the reproducing kernel. Also in the second limit we recover standard results obeyed by Hermite polynomials in $z$ and $\bar{z}$.

math.CA↗
arXiv2018

Coherence, squeezing and entanglement -- an example of peaceful coexistence

After exhaustive inspection of bosonic coherent states appearing in physical literature two of us, Horzela and Szafraniec, came in 2012 to the reasonably general definition which relies exclusively on reproducing kernels. The basic feature of coherent states, which is the resolution of the identity, is preserved though it now achieves advantageous form of the Segal-Bargmann transform. It turns out that the aforesaid definition is not only extremely economical but also puts under a common umbrella typical coherent states as well as those which are squeezed and entangled. We examine the case here on the groundwork of holomorphic Hermite polynomials in one and two variables. An interesting side of this story is how some limit procedure allows disentangling.

math-ph↗
arXiv2017

Mittag-Leffler function and fractional differential equations

We adopt a procedure of operational-umbral type to solve the $(1+1)$-dimensional fractional Fokker-Planck equation in which time fractional derivative of order $α$ ($0 < α< 1$) is in the Riemann-Liouville sense. The technique we propose merges well documented operational methods to solve ordinary FP equation and a redefinition of the time by means of an umbral operator. We show that the proposed method allows significant progress including the handling of operator ordering.

math-ph↗
arXiv2017

The Havriliak-Negami relaxation and its relatives: the response, relaxation and probability density functions

We study functions related to the experimentally observed Havriliak-Negami dielectric relaxation pattern in the frequency domain $\sim[1+(iωτ_{0})^α]^{-β}$ with $τ_{0}$ being some characteristic time. For $α= l/k< 1$ ($l$ and $k$ positive integers) and $β> 0$ we furnish exact and explicit expressions for response and relaxation functions in the time domain and suitable probability densities in their "dual" domain. All these functions are expressed as finite sums of generalized hypergeometric functions, convenient to handle analytically and numerically. Introducing a reparameterization $β= (2-q)/(q-1)$ and $τ_{0} = (q-1)^{1/α}$ $(1 < q < 2)$ we show that for $0 < α< 1$ the response functions $f_{α, β}(t/τ_{0})$ go to the one-sided Lévy stable distributions when $q$ tends to one. Moreover, applying the self-similarity property of the probability densities $g_{α, β}(u)$, we introduce two-variable densities and show that they satisfy the integral form of the evolution equation.

cond-mat.stat-mech↗
arXiv2017

The stretched exponential behavior and its underlying dynamics. The phenomenological approach

We show that the anomalous diffusion equations with a fractional derivative in the Caputo or Riesz sense are strictly related to the special convolution properties of the Lévy stable distributions which stem from the evolution properties of stretched or compressed exponential function. The formal solutions of these fractional differential equations are found by using the evolution operator method where the evolution operator is presented as integral transforms whose kernel is the Green function. Exact and explicit examples of the solutions are reported and studied for various fractional order of derivatives and different initial conditions.

cond-mat.stat-mech↗
arXiv2016

Theory of relativistic heat polynomials and one-sided Lévy distributions

The theory of pseudo-differential operators is a powerful tool to deal with differential equations involving differential operators under the square root sign. These type of equations are pivotal elements to treat problems in anomalous diffusion and in relativistic quantum mechanics. In this paper we report on new and unsuspected links between fractional diffusion, quantum relativistic equations and particular families of polynomials, linked to the Carlitz family, and playing the role of relativistic heat polynomials. We introduce generalizations of these polynomial families and point out their specific use for the solutions of problems of practical importance.

math-ph↗
arXiv2016

Relativistic Heat Equation via Lévy stable distributions: Exact Solutions

We introduce and study an extension of the heat equation relevant to relativistic energy formula involving square root of differential operators. We furnish exact solutions of corresponding Cauchy (initial) problem using the operator formalism invoking one-sided Lévy stable distributions. We note a natural appearance of Bessel polynomials which allow one the obtention of closed form solutions for a number of initial conditions. The resulting relativistic diffusion is slower than the non-relativistic one, although it still can be termed a normal one. Its detailed statistical characterization is presented in terms of exact evaluation of arbitrary moments and is compared with the non-relativistic case.

math-ph↗
arXiv2016

On the properties of Laplace transform originating from one-sided Lévy stable laws

We consider the conventional Laplace transform of $f(x)$, denoted by $\mathcal{L}[f(x); p]~\equiv~F(p)=\int_{0}^{\infty} e^{-p x} f(x) dx$ with ${\rm \mathfrak{Re}}(p) > 0$. For $0 < α< 1$ we furnish the closed form expressions for the inverse Laplace transforms $\mathcal{L}^{-1}[F(p^α); x]$ and $\mathcal{L}^{-1}[p^{α-1}F(p^α); x]$. In both cases they involve definite integration with kernels which are appropriately rescaled one-sided Lévy stable probability distribution functions $g_α(x)$, $0 < α< 1$, $x > 0$. Since $g_α(x)$ are exactly and explicitly known for rational $α$, \textit{i.e.} for $α= l/k$ with $l, k=1, 2, \ldots$, $l < k$, our results extend the known and tabulated case of $α= 1/2$ to any rational $0 < α< 1$. We examine the integral kernels of this procedure as well as the resulting two kinds of Lévy integral transformations.

math-ph↗
arXiv2015

Relativistic Wave Equations: An Operational Approach

The use of operator methods of algebraic nature is shown to be a very powerful tool to deal with different forms of relativistic wave equations. The methods provide either exact or approximate solutions for various forms of differential equations, such as relativistic Schrödinger, Klein-Gordon and Dirac. We discuss the free particle hypotheses and those relevant to particles subject to non-trivial potentials. In the latter case we will show how the proposed method leads to easily implementable numerical algorithms.

math-ph↗
arXiv2015

Explicit representations for multiscale Lévy processes, and asymptotics of multifractal conservation laws

Nonlinear conservation laws driven by Lévy processes have solutions which, in the case of supercritical nonlinearities, have an asymptotic behavior dictated by the solutions of the linearized equations. Thus the explicit representation of the latter is of interest in the nonlinear theory. In this paper we concentrate on the case where the driving Lévy process is a multiscale stable (anomalous) diffusion, which corresponds to the case of multifractal conservation laws considered in [1-4]. The explicit representations, building on the previous work on single-scale problems (see, e.g.,[5]), are developed in terms of the special functions (such as Meijer G functions), and are amenable to direct numerical evaluations of relevant probabilities.

math-ph↗
arXiv2014

On the Laplace transform of the Fréchet distribution

We calculate exactly the Laplace transform of the Fréchet distribution in the form $γx^{-(1+γ)} \exp(-x^{-γ})$, $γ> 0$, $0 \leq x < \infty$, for arbitrary rational values of the shape parameter $γ$, i.e. for $γ= l/k$ 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 Fréchet distribution serves as a kernel of Fréchet integral transform. It turns out that the Fréchet transform of one-sided Lévy law reproduces the Fréchet distribution.

math.PR↗
arXiv2012

Definite integrals and operational methods

An operatorial method, already employed to formulate a generalization of the Ramanujan master theorem, is applied to the evaluation of integrals of various type. This technique provide a very flexible and powerful tool yielding new results encompassing various aspects of the special function theory.

math.CA↗
arXiv2012

On Mittag-Leffler function and associated polynomials

The Mittag-Leffler function plays a role of central importance in the theory of fractional derivatives. In this brief note we discuss the properties of this function and its connection with the Wright-Bessel functions and with a new family of associated heat polynomials.

math-ph↗
arXiv2011

Repeated derivatives of composite functions and generalizations of the Leibniz rule

We use the properties of Hermite and Kampé de Fériet polynomials to get closed forms for the repeated derivatives of functions whose argument is a quadratic or higher-order polynomial. The results we obtain are extended to product of functions of the above argument, thus giving rise to expressions which can formally be interpreted as generalizations of the familiar Leibniz rule. Finally, examples of practical interest are discussed.

math.CA↗