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We provide a review of theoretical results concerning the Havriliak–Negami (HN) and the Jurlewicz–Weron–Stanislavsky (JWS) dielectric relaxation models. We derive explicit forms of functions characterizing relaxation phenomena in the time domain—the relaxation, response and probability distribution functions. We also explain how to construct and solve relevant evolution equations within these models. These equations are usually solved by using the Schwinger parametrization and the integral transforms. Instead, in this work we replace it by the powerful Efros theorem. That allows one to relate physically admissible solutions to the memory-dependent evolution equations with phenomenologically known spectral functions and, from the other side, with the subordination mechanism emerging from a stochastic analysis of processes underpinning considered relaxation phenomena. Our approach is based on a systematic analysis of the memory-dependent evolution equations. It exploits methods of integral transforms, operational calculus and special functions theory with the completely monotone and Bernstein functions. Merging analytic and stochastic methods enables us to give a complete classification of the standard functions used to describe the large class of the relaxation phenomena and to explain their properties. [ABSTRACT FROM AUTHOR]

The paper studies the exact solution of two kinds of generalized Fokker–Planck equations in which the integral kernels are given either by the distributed order function k 1 (t) = ∫ 0 1 t − μ / Γ (1 − μ) d μ or the distributed order Prabhakar function k 2 (α , γ ; λ ; t) = ∫ 0 1 e α , 1 − μ − γ (λ ; t) d μ , where the Prabhakar function is denoted as e α , 1 − μ − γ (λ ; t). Both of these integral kernels can be called the fading memory functions and are the Stieltjes functions. It is also shown that their Stieltjes character is enough to ensure the non-negativity of the mean square values and higher even moments. The odd moments vanish. Thus, the solution of generalized Fokker–Planck equations can be called the probability density functions. We introduce also the Volterra-Prabhakar function and its generalization which are involved in the definition of k 2 (α , γ ; λ ; t) and generated by it the probability density function p 2 (x , t). [ABSTRACT FROM AUTHOR]

Highlights • In contradiction to the Debye relaxation the time composition of non-Debye relaxation processes can not be given by multiplication and to preserve the causality must be replaced by some more general rule. • We construct explicitly an example of such a rule for the Cole-Cole relaxation described in the time domain by the Mittag-Leffler function. • The time composition law for the Cole-Cole relaxation leads to the time evolution equations involving fractional derivatives. • It provides us with a meaningful argument that the natural language to used in investigations of processes with memory is that of fractional calculus. • The method used to represent the time evolution law for the Cole-Cole relaxation suggests how to analyze and to solve analogous problem for more general models of relaxation processes, e.g. the Havriliak-Negami model, and how to find suitable evolution equations. Abstract Physically natural assumption says that any relaxation process taking place in the time interval [ t 0 , t 2 ] , t 2 > t 0 ≥ 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 ≤ t 1 ≤ 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. [ABSTRACT FROM AUTHOR]

Abstract 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 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" behaviour 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 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 z ‾. [ABSTRACT FROM AUTHOR]

The theory of pseudo-differential operators is a powerful tool to deal with differential equations involving differential operators under the square root sign. These types of equations are pivotal elements to treat problems in anomalous diffusion and in relativistic quantum mechanics. In this paper, we report on new links between fractional diffusion, quantum relativistic equations, and particular families of polynomials, linked to the Bessel polynomials in Carlitz form 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. [ABSTRACT FROM AUTHOR]

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 Biler et al. [J. Differ. Equations 148, 9 (1998); Stud. Math. 135, 231 (1999); Ann. Inst. Henri Poincare, Anal. Nonlineaire 18, 613 (2001); and Stud. Math. 148, 171 (2001)]. The explicit representations, building on the previous work on single-scale problems (see, e.g., Górska and Penson [Phys. Rev. E 83, 061125 (2011)]), are developed in terms of the special functions (such as Meijer G functions) and are amenable to direct numerical evaluations of relevant probabilities. [ABSTRACT FROM AUTHOR]

We calculate exactly the Laplace transform of the Fréchet distribution in the form γx-(1 + γ)exp (-x-γ), γ > 0, 0 ≤ x < ∞, for arbitrary rational values of the shape parameter γ, i.e. for γ = l/k with l, k = 1, 2, . . . . 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. [ABSTRACT FROM AUTHOR]

We incorporate the tools of Lévy processes and distributions to describe the photoluminescence of silicon nanocrystals. The method relies on two novel features: first we use exact forms of one-sided Lévy distributions to get an excellent reproduction of experimental data. Then we show that the dynamics leading to photoluminescence decay can be modelled in terms of fractional Fokker-Planck equation. [ABSTRACT FROM AUTHOR]

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. Pure Appl. Math. 14, 187 (1961)]. [ABSTRACT FROM AUTHOR]

We study the higher-order heat-type equation with first time and Mth spatial partial derivatives, M = 2, 3, ... . We demonstrate that its exact solutions for M even can be constructed with the help of signed Lévy stable functions. For M odd the same role is played by a special generalization of the 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 spatial and temporary evolution of particular solutions for simple initial conditions. [ABSTRACT FROM AUTHOR]

Abstract: We study a number of possible extensions of the Ramanujan master theorem, which is formulated here by using methods of Umbral nature. We discuss the implications of the procedure for the theory of special functions, like the derivation of formulae concerning the integrals of products of families of Bessel functions and the successive derivatives of Bessel type functions. We stress also that the procedure we propose allows a unified treatment of many problems appearing in applications, which can formally be reduced to the evaluation of exponential- or Gaussian-like integrals. [Copyright &y& Elsevier]

We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Lévy stable probability distributions gα(x), 0 <= x < ∞, 0 < α < 1. We demonstrate that the knowledge of one such a distribution gα(x) suffices to obtain exactly gαp(x), p = 2, 3, ... . Similarly, from known gα(x) and gβ(x), 0 < α, β < 1, we obtain gαβ(x). The method is based on the construction of the integral operator, called Lévy transform, which implements the above operations. For α rational, α = l/k with l < k, we reproduce in this manner many of the recently obtained exact results for gl/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. [ABSTRACT FROM AUTHOR]

We obtain exact results for fractional equations of Fokker-Planck type using the evolution operator method. We employ exact forms of one-sided Lévy stable distributions to generate a set of self-reproducing solutions. Explicit cases are reported and studied for various fractional order of derivatives, different initial conditions, and for different versions of Fokker-Planck operators. [ABSTRACT FROM AUTHOR]

We investigate functions g(α,β;x) which are heavy-tailed Lévy stable probability distributions of index 0<α⩽2 and appropriate asymmetry parameter β. They are of central importance in physics of amorphous and disordered systems, econophysics, geology, hydrology, internet traffic, dynamics of human relations, etc. We present an ensemble of exact and explicit solutions of g(α,β;x) for all admissible rational values of α and β. We reproduce all the previously known cases and furnish distributions for values α and β which were previously inaccessible. We point out many instances of experimental and statistical data that could be described by our results. [ABSTRACT FROM AUTHOR]

We consider discrete spectra of bound states for nonrelativistic motion in attractive potentials Vσ(x)=-|V0| |x|-σ, 0<σ≤2. For these potentials the quasiclassical approximation for n → ∞ predicts quantized energy levels eσ(n) of a bounded spectrum varying as eσ(n)∼-n-2σ/(2-σ). We construct collective quantum states using the set of wavefunctions of the discrete spectrum assuming this asymptotic behavior. 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 σ in the range 0 < σ <= 2/3 we present exact implementations of such states for the parametrization σ = 2(k - l)/(3k - l) with k and l positive integers satisfying k>l. [ABSTRACT FROM AUTHOR]

We studied the relationships between crop yield variability (imposed mainly by the weather) and the scale of area aggregation. Empirical data comprised yields of cultivated plants in Poland in the 1956–1990 period. Starting from the general formula for calculating the variance of a sum, the yield variance in an aggregated area was described as a complex function of variances in individual area units and of correlations between them. Since the coefficient of correlation may be presented as a function of distance between fields, it was possible to construct a general formula for assessment of yield variability in areas of any desired size:ln V=ln v – bn-10.2where: V, coefficient of yield variability in an aggregated area; v, in a unit area; n, number of unit areas; b, empirical parameter. The parameters for several crops in Poland are presented. [Copyright &y& Elsevier]

The paper is devoted to study the frequency shift in the solution of the generalized telegraph equation with a moving point-wise harmonic source. This equation contains the nonlocality in time derivatives which is expressed by the memory functions η (t) and γ (t) , where η (t) smears the second time-derivative and γ (t) the first one. Moreover, in the Laplace domain we have η ˆ (s) = γ ˆ 2 (s). The generalized telegraph equation with an external source is solved by using the integral decomposition which allows us to write this solution as a product of the solution of the telegrapher equation with harmonic source and f γ ˆ (ξ , t) which is a function of the Laplace transform of memory function γ. Such obtained solution manifests the frequency shift which is illustrated in three examples of the memory functions γ (t) : the localized case, its mixture with power-law, and the power-law case only. We show that only the first two cases have the wave front and the Doppler-like shift. The third example, despite the lack of wave fronts, also manifests the frequency shift. Thus it turns out that the frequency shift occurs regardless of the existence of a wave front, but it is more visible when such a front exists. • The method of integral decomposition is applied to get solutions of the memory dependent evolution equations. • Jugging existence/non-existence of the Doppler shift is separated from the existence/non-existence the wave fronts. • It is shown that the Cattaneo-Vernotte equation may be used to describe the wave properties of the heat transport. [ABSTRACT FROM AUTHOR]

Inspired by the article "Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel" (Z. Angew. Math. Phys. (2019) 70:42) whose authors Zhao and Sun studied the integro-differential equation with the kernel given by the Prabhakar function e α , β - γ (t , λ) , we provide the solution to this equation which is complementary to that obtained up to now. Our solution is valid for effective relaxation times whose admissible range extends the limits given in Zhao and Sun (Z Angew Math Phys 70:42, 2019, Theorem 3.1) to all positive values. For special choices of parameters entering the equation itself and/or characterizing the kernel, the solution comprises to known phenomenological relaxation patterns, e.g., to the Cole–Cole model (if γ = 1 , β = 1 - α ) or to the standard Debye relaxation. [ABSTRACT FROM AUTHOR]

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. These results 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. [ABSTRACT FROM AUTHOR]

Abstract: 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 successive 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. [Copyright &y& Elsevier]

The use of the umbral formalism allows a significant simplification of the derivation of sum rules involving products of special functions and polynomials. We rederive in this way known sum rules and addition theorems for Bessel functions. Furthermore, we obtain a set of new closed form sum rules involving various special polynomials and Bessel functions. The examples we consider are relevant for applications ranging from plasma physics to quantum optics. [ABSTRACT FROM AUTHOR]

Abstract: An operational method, already employed to formulate a generalization of the Ramanujan master theorem, is applied to the evaluation of integrals of various types. This technique provides a very flexible and powerful tool yielding new results encompassing different aspects of the special function theory. [Copyright &y& Elsevier]

This paper completes our study of coherent states in the so-called magnetic- solenoid field (a collinear combination of a constant uniform magnetic field and Aharonov-Bohm solenoid field) presented in Bagrov et al (2010 J. Phys. A: Math. Theor. 43 354016, 2011 J. Phys. A: Math. Theor. 44 055301). Here, we succeeded in proving nontrivial completeness relations for non-relativistic and relativistic coherent states in such a field. In addition, we solve here the relevant Stieltjes moment problem and present a comparative analysis of our coherent states and the well-known, in the case of pure uniform magnetic field, Malkin-Man'ko coherent states. [ABSTRACT FROM AUTHOR]