Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

6 Issues per year

IMPACT FACTOR 2017: 2.865
5-year IMPACT FACTOR: 3.323

CiteScore 2017: 3.06

SCImago Journal Rank (SJR) 2017: 1.967
Source Normalized Impact per Paper (SNIP) 2017: 1.954

Mathematical Citation Quotient (MCQ) 2017: 0.98

See all formats and pricing
More options …

Generalization of the fractional poisson distribution

Richard Herrmann
Published Online: 2016-08-27 | DOI: https://doi.org/10.1515/fca-2016-0045


A generalization of the Poisson distribution based on the generalized Mittag-Leffler function Eα,β(λ) is proposed and the raw moments are calculated algebraically in terms of Bell polynomials. It is demonstrated, that the proposed distribution function contains the standard fractional Poisson distribution as a subset. A possible interpretation of the additional parameter β is suggested.

MSC 2010: Primary 26A33; Secondary 33E12, 60EXX, 11B73

Key Words and Phrases: fractional calculus; Mittag-Leffler functions; fractional Poisson distribution; Bell polynomials; Stirling numbers


  • [1]

    E. T. Bell, Exponential polynomials. Ann. Math. 35, No 2 (1934), 258–277.Google Scholar

  • [2]

    S. Chakraborty, S. H. Ong, Mittag-Leffler function distribution - A new generalization of hyper-Poisson distribution. arXiv:1411.0980 [math.ST] (2014).Google Scholar

  • [3]

    G. Dobinski, Summirung der Reihe Σ nm/n! für m = 1,2,3,4,5, ... . Grunert Archiv (Arch. Math. Phys.) 61 (1877), 333–336.Google Scholar

  • [4]

    R. Garra, E. Orsingher, Random flights governed by Klein-Gordon-type partial differential equations. Stoch. Proc. Appl. 124 (2014), 2171–2187; .CrossrefGoogle Scholar

  • [5]

    R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014).Google Scholar

  • [6]

    R. Gorenflo, F. Mainardi, On the fractional Poisson process and the discretized stable subordinator. Axioms 4 (2015), 321–344; .CrossrefGoogle Scholar

  • [7]

    H. J. Haubold, A. M. Mathai, R. K. Saxena, Mittag-Leffler functions and their applications. Journal of Applied Mathemathics 2011 (2011), Article ID 298628; .CrossrefGoogle Scholar

  • [8]

    A. A. Kilbas, A. A. Koroleva, S. S. Rogosin, Multi-parameter Mittag-Leffler functions and their extension. Fract. Calc. Appl. Anal. 16, No 2 (2013), 378–404; ; http://www.degruyter.com/view/j/fca.2013.16.issue-2/issue-files/fca.2013.16.issue-2.xml.CrossrefGoogle Scholar

  • [9]

    V. Kiryakova, Multi-indexed Mittag-Leffler functions, related Gelfond-Leontiev operators and Laplace type transforms. Fract. Calc. Appl. Anal. 2, No 4 (1999), 445–462.Google Scholar

  • [10]

    N. Laskin, Fractional Poisson process. Commun. Nonlin. Sci. Num. Sim. 8 (2003), 201–213; .CrossrefGoogle Scholar

  • [11]

    N. Laskin, Some applications of the fractional Poisson probability distribution. J. Math. Phys. 50 (2009), 113513; .CrossrefGoogle Scholar

  • [12]

    F. Mainardi, R. Gorenflo, E. Scalas, A fractional generalization of the Poisson processes. Vietnam Journal of Mathematics 32, SI (2004), 53–64; E-print http://arxiv.org/abs/math/0701454.Google Scholar

  • [13]

    M. M. Meerschaert, D. A. Benson, B. Bäumer, Multidimensional advection and fractional dispersion. Phys. Rev. E 59 (1999), 5026; .CrossrefGoogle Scholar

  • [14]

    M. M. Meerschaert, E. Nane, P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator, Electronic Journal of Probability 16, No 59 (2011), 1600–1620; see also arXiv:1007.5051[math.PR].Google Scholar

  • [15]

    R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77; .CrossrefGoogle Scholar

  • [16]

    M. G. Mittag-Leffler, Sur la nouvelle function Eα(x). Comptes Rendus Acad. Sci. Paris 137 (1903), 554–558.Google Scholar

  • [17]

    J. D. Murray, Mathematical Biology I: An Introduction. 3th Ed., Springer, Berlin (2008).Google Scholar

  • [18]

    I. Podlubny, Fractional Differential Equations. Academic Press, Boston (1999).Google Scholar

  • [19]

    M. Politi, T. Kaizoji, E. Scalas, Full characterization of the fractional Poisson process. EPL 96 (2011), 20004; .CrossrefGoogle Scholar

  • [20]

    O. N. Repin, A. I. Saichev, Fractional Poisson law. Radiophys. Quant. Electron. 43 (2000), 738–741; .CrossrefGoogle Scholar

  • [21]

    S. Roman, ”The Exponential Polynomials” and ”The Bell Polynomials”, 4.1.3 and 4.1.8. In: The Umbral Calculus. Academic Press, New York (1984), 63–67 and 82–87.Google Scholar

  • [22]

    S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach, Yverdon (1993); Transl. and extended from the 1987 Russian original.Google Scholar

  • [23]

    J. M. Sixdeniers, K. A. Penson, A. I. Solomon, Mittag-Leffler coherent states. J. Phys. A.: Math. Gen. 32 (1999), 7543; .CrossrefGoogle Scholar

  • [24]

    J. Stirling, Methodus differentialis, Sive tractatus de summatione et interpolatione serierum infinitarium, London (1730); English transl. by J. Holliday, The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series (1749).Google Scholar

  • [25]

    V. V. Uchaikin, D. O. Cahoy, R. T. Sibatov, Fractional processes: from Poisson to branching one. Int. J. Bifurcation Chaos 18, No 9 (2008), 2717–2725; ; arXiv:1002.2511v1.CrossrefGoogle Scholar

  • [26]

    G. C. Wick, The evaluation of the collision matrix. Phys. Rev. 80 (1950), 268; .CrossrefGoogle Scholar

  • [27]

    A. Wiman, Über den Fundamentalsatz in der Theorie der Funktionen Eα(x). Acta Math. 29 (1905), 191–201; .CrossrefGoogle Scholar

About the article

Received: 2015-03-10

Published Online: 2016-08-27

Published in Print: 2016-08-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 19, Issue 4, Pages 832–842, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2016-0045.

Export Citation

© 2016 Diogenes Co., Sofia.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Giampietro Casasanta and Roberto Garra
Fractal and Fractional, 2018, Volume 2, Number 1, Page 8
Roberto Garra, Enzo Orsingher, and Federico Polito
Mathematics, 2018, Volume 6, Number 1, Page 4
Tibor K. Pogány and Živorad Tomovski
Integral Transforms and Special Functions, 2016, Volume 27, Number 10, Page 783

Comments (0)

Please log in or register to comment.
Log in