The H-Function

Mathai, A. M.; Saxena, R. K.; Haubold, H. J.

doi:10.1007/978-1-4419-0916-9

Cited by 481 publications

(171 citation statements)

References 3 publications

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“…However, these brackets can be exchanged due to property 1.1 of the H-function in Ref. [60]. Thus, the H-function solution for the unbiased case (v = 0) is obtained correctly.…”

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confidence: 83%

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Space-fractional Fokker–Planck equation and optimization of random search processes in the presence of an external bias

2014

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We study the efficiency of random search processes based on Lévy flights with power-law distributed jump lengths in the presence of an external drift, for instance, an underwater current, an airflow, or simply the bias of the searcher based on prior experience. While Lévy flights turn out to be efficient search processes when relative to the starting point the target is upstream, in the downstream scenario regular Brownian motion turns out to be advantageous. This is caused by the occurrence of leapovers of Lévy flights, due to which Lévy flights typically overshoot a point or small interval. Extending our recent work on biased LF search [V. V. Palyulin, A. V. Chechkin, and R. Metzler, Proc. Natl. Acad. Sci. USA, 111, 2931 (2014).] we establish criteria when the combination of the external stream and the initial distance between the starting point and the target favors Lévy flights over regular Brownian search. Contrary to the common belief that Lévy flights with a Lévy index α = 1 (i.e., Cauchy flights) are optimal for sparse targets, we find that the optimal value for α may range in the entire interval (1, 2) and include Brownian motion as the overall most efficient search strategy.

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“…However, these brackets can be exchanged due to property 1.1 of the H-function in Ref. [60]. Thus, the H-function solution for the unbiased case (v = 0) is obtained correctly.…”

mentioning

confidence: 83%

“…Then for α = 2 the order of H-function is reduced by use of the properties 1.2 and 1.3 from chapter 1 in Ref. [60]) as well as its definition via the Mellin transform [60]. This procedure yields…”

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confidence: 99%

Space-fractional Fokker–Planck equation and optimization of random search processes in the presence of an external bias

2014

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“…The H-function reduces to a large number of special function [7,9,10,13,23] and so from Theorems 1-4, we can further obtain various fractional calculus results involving a number of special function. Here, we provide a few special cases of our main findings (see also, [1,3,24,29,30]).…”

Section: Special Casesmentioning

confidence: 99%

FRACTIONAL CALCULUS FORMULAS INVOLVING <TEX>$\bar{H}$</TEX>-FUNCTION AND SRIVASTAVA POLYNOMIALS

Kumar¹

2016

Communications of the Korean Mathematical Society

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Abstract. Here, in this paper, we aim at establishing some new unified integral and differential formulas associated with the H-function. Each of these formula involves a product of the H-function and Srivastava polynomials with essentially arbitrary coefficients and the results are obtained in terms of two variables H-function. By assigning suitably special values to these coefficients, the main results can be reduced to the corresponding integral formulas involving the classical orthogonal polynomials including, for example, Hermite, Jacobi, Legendre and Laguerre polynomials. Furthermore, the H-function occurring in each of main results can be reduced, under various special cases.

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“…These functions, introduced by Fox in 1961 [33], are special functions of a very general nature which allow us to present the results in a universal and elegant fashion. For a general theory on the H−functions we address the reader to the monograph of Mathai and Saxena [34] and to ref. [35].…”

Section: Fractional Gaussian Noise Correlation Functionmentioning

confidence: 99%

“…This list is not an exhaustive compendium of the Fox functions properties, for which the reader could refer to [34,35,41]. For convenience in this Section we adopt the following short notation…”

Section: Appendix C: Fox Function Propertiesmentioning

confidence: 99%

Correlations in a generalized elastic model: Fractional Langevin equation approach

2010

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The Generalized Elastic Model (GEM) provides the evolution equation which governs the stochastic motion of several many-body systems in nature, such as polymers, membranes, growing interfaces. On the other hand a probe (tracer ) particle in these systems performs a fractional Brownian motion due to the spatial interactions with the other system's components. The tracer's anomalous dynamics can be described by a Fractional Langevin Equation (FLE) with a space-time correlated noise. We demonstrate that the description given in terms of GEM coincides with that furnished by the relative FLE, by showing that the correlation functions of the stochastic field obtained within the FLE framework agree to the corresponding quantities calculated from the GEM. Furthermore we show that the Fox H-function formalism appears to be very convenient to describe the correlation properties within the FLE approach.

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The H-Function

Cited by 481 publications

References 3 publications

Space-fractional Fokker–Planck equation and optimization of random search processes in the presence of an external bias

Space-fractional Fokker–Planck equation and optimization of random search processes in the presence of an external bias

FRACTIONAL CALCULUS FORMULAS INVOLVING <TEX>$\bar{H}$</TEX>-FUNCTION AND SRIVASTAVA POLYNOMIALS

Correlations in a generalized elastic model: Fractional Langevin equation approach

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