On Mittag-Leffler functions and related distributions

Pillai, R. N.

doi:10.1007/bf00050786

Cited by 231 publications

(151 citation statements)

References 3 publications

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“…Pillai [31] who showed that these distributions are infinitely divisible and geometrically infinitely divisible for ρ ∈ (0, 1]. For our purposes, it is especially relevant that the Mittag-Leffler distribution has Laplace transform…”

Section: Scaling Solutions For the Constant Kernel 41 Mittag-lefflermentioning

confidence: 99%

“…For ρ = 1 these solutions reduce to the known solutions with exponential tails, while for 0 < ρ < 1 the number density has algebraic decay ("fat tails"). For K = 2 (γ = 0) the normalized size distribution is a Mittag-Leffler distribution as studied by Pillai [31]. For K = x + y (γ = 1) and xy (γ = 2) the γ-th moment distributions are transformed by power-law rescaling to the Lévy stable laws of probability theory (see 6.5 and [7]).…”

Section: Introductionmentioning

confidence: 99%

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Approach to self‐similarity in Smoluchowski's coagulation equations

Menon

Pego

2003

Comm Pure Appl Math

129

293

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We consider the approach to self-similarity (or dynamical scaling) in Smoluchowski's equations of coagulation for the solvable kernels K(x, y) = 2, x + y and xy. In addition to the known self-similar solutions with exponential tails, there are one-parameter families of solutions with algebraic decay, whose form is related to heavy-tailed distributions well-known in probability theory. For K = 2 the size distribution is Mittag-Leffler, and for K = x + y and K = xy it is a power-law rescaling of a maximally skewed α-stable Lévy distribution. We characterize completely the domains of attraction of all self-similar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits.

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Section: Scaling Solutions For the Constant Kernel 41 Mittag-lefflermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approach to self‐similarity in Smoluchowski's coagulation equations

Menon

Pego

2003

Comm Pure Appl Math

129

293

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“…A statistical density in terms of Mittag-Leffler function was originally defined by Pillai [38], Pillai and Jayakumar [39], in terms of the following distribution function or cumulative density:…”

Section: A Generalized Mittag-leffler Statistical Densitymentioning

confidence: 99%

Matrix-variate statistical distributions and fractional calculus

Mathai

Haubold

2011

fcaa

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Dedicated to Professor R. Gorenflo on the occasion of his 80th birthdayA connection between fractional calculus and statistical distribution theory has been established by the authors recently. Some extensions of the results to matrix-variate functions were also considered. In the present article, more results on matrix-variate statistical densities and their connections to fractional calculus will be established. When considering solutions of fractional differential equations, Mittag-Leffler functions and Fox Hfunction appear naturally. Some results connected with generalized MittagLeffler density and their asymptotic behavior will be considered. Reference is made to applications in physics, particularly superstatistics and nonextensive statistical mechanics.MSC 2010 : 15A15, 15A52, 33C60, 33E12, 44A20, 62E15

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“…In general, the derived form (2.5) corresponds to the whole class of response functions observed in dielectric relaxation phenomena. The eective relaxation rateb Ã leading to (2.5) is distributed according to the negative-binomial-stable law that can be recognized as the generalized Mittag±Leer distribution [38] …”

Section: Survival Probability As the ®Rst Passage Of A Complex Systemmentioning

confidence: 99%

Relaxation dynamics of the fastest channel in multichannel parallel relaxation mechanism

Jurlewicz

Weron

2000

Chaos, Solitons & Fractals

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Empirical evidence accumulated over the years shows that the time-dependent change of macroscopic properties of physical systems evolving to equilibrium exhibits a great degree of universality. We address the question of the origins of the universal relaxation laws in terms of probabilistic approach based on the multichannel parallel relaxation mechanism. We present a clear stochastic scheme uniquely leading to the whole class of experimentally observed relaxation responses. The proposed scheme results from the common assumption that only the fastest channel contributes to relaxation dynamics. Ó

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On Mittag-Leffler functions and related distributions

Abstract: Completely monotone function, Laplace transform, infinite divisibility, geometric infinite divisibility, stable process,

Cited by 231 publications

References 3 publications

Approach to self‐similarity in Smoluchowski's coagulation equations

Approach to self‐similarity in Smoluchowski's coagulation equations

Matrix-variate statistical distributions and fractional calculus

Relaxation dynamics of the fastest channel in multichannel parallel relaxation mechanism

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