Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter August 27, 2016

Generalization of the fractional poisson distribution

  • Richard Herrmann EMAIL logo


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.


We thank A. Friedrich for valuable discussions.


[1] E. T. Bell, Exponential polynomials. Ann. Math. 35, No 2 (1934), 258–277.10.2307/1968431Search in 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).10.1186/s40488-017-0060-9Search in 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.Search in Google Scholar

[4] R. Garra, E. Orsingher, Random flights governed by Klein-Gordon-type partial differential equations. Stoch. Proc. Appl. 124 (2014), 2171–2187; 10.1016/ in Google Scholar

[5] R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014).10.1007/978-3-662-43930-2Search in Google Scholar

[6] R. Gorenflo, F. Mainardi, On the fractional Poisson process and the discretized stable subordinator. Axioms4 (2015), 321–344; 10.3390/axioms4030321.Search in Google Scholar

[7] H. J. Haubold, A. M. Mathai, R. K. Saxena, Mittag-Leffler functions and their applications. Journal of Applied Mathemathics2011 (2011), Article ID 298628; 10.1155/2011/298628.Search in Google 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; 10.2478/s13540-013-0024-9; in Google 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.Search in Google Scholar

[10] N. Laskin, Fractional Poisson process. Commun. Nonlin. Sci. Num. Sim. 8 (2003), 201–213; 10.1016/S1007-5704(03)00037-6.Search in Google Scholar

[11] N. Laskin, Some applications of the fractional Poisson probability distribution. J. Math. Phys. 50 (2009), 113513; /10.1063/1.3255535.Search in Google Scholar

[12] F. Mainardi, R. Gorenflo, E. Scalas, A fractional generalization of the Poisson processes. Vietnam Journal of Mathematics32, SI (2004), 53–64; E-print in Google Scholar

[13] M. M. Meerschaert, D. A. Benson, B. Bäumer, Multidimensional advection and fractional dispersion. Phys. Rev. E59 (1999), 5026; 10.1103/PhysRevE.59.5026.Search in Google Scholar PubMed

[14] M. M. Meerschaert, E. Nane, P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator, Electronic Journal of Probability16, No 59 (2011), 1600–1620; see also arXiv:1007.5051[math.PR].10.1214/EJP.v16-920Search in 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; 10.1016/S0370-1573(00)00070-3.Search in Google Scholar

[16] M. G. Mittag-Leffler, Sur la nouvelle function Eα(x). Comptes Rendus Acad. Sci. Paris137 (1903), 554–558.Search in Google Scholar

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

[18] I. Podlubny, Fractional Differential Equations. Academic Press, Boston (1999).Search in Google Scholar

[19] M. Politi, T. Kaizoji, E. Scalas, Full characterization of the fractional Poisson process. EPL96 (2011), 20004; 10.1209/0295-5075/96/20004.Search in Google Scholar

[20] O. N. Repin, A. I. Saichev, Fractional Poisson law. Radiophys. Quant. Electron. 43 (2000), 738–741; 10.1023/A:1004890226863.Search in Google 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.Search in 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.Search in Google Scholar

[23] J. M. Sixdeniers, K. A. Penson, A. I. Solomon, Mittag-Leffler coherent states. J. Phys. A.: Math. Gen. 32 (1999), 7543; 10.1088/0305-4470/32/43/308.Search in Google 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).Search in Google Scholar

[25] V. V. Uchaikin, D. O. Cahoy, R. T. Sibatov, Fractional processes: from Poisson to branching one. Int. J. Bifurcation Chaos18, No 9 (2008), 2717–2725; 10.1142/S0218127408021932; arXiv:1002.2511v1.Search in Google Scholar

[26] G. C. Wick, The evaluation of the collision matrix. Phys. Rev. 80 (1950), 268; 10.1103/PhysRev.80.268.Search in Google Scholar

[27] A. Wiman, Über den Fundamentalsatz in der Theorie der Funktionen Eα(x). Acta Math. 29 (1905), 191–201; 10.1007/BF02403202.Search in Google Scholar

Received: 2015-3-10
Published Online: 2016-8-27
Published in Print: 2016-8-1

© 2016 Diogenes Co., Sofia

Downloaded on 22.2.2024 from
Scroll to top button