Generalization of the fractional Poisson distribution
aa r X i v : . [ m a t h . S T ] M a r GENERALIZATION OF THE FRACTIONAL POISSONDISTRIBUTIONRichard Herrmann Abstract
A generalization of the Poisson distribution based on the generalizedMittag-Leffler function E α,β ( λ ) is proposed and the raw moments are cal-culated algebraically in terms of Bell polynomials. It is demonstrated, thatthe proposed distribution function contains the standard fractional Pois-son distribution as a subset. A possible interpretation of the additionalparameter 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. Introduction
Since its early beginnings, fractional calculus has developed as an evergrowing powerful tool to model complex phenomena in different branchesof e.g. physics or biology.A major field of research is the area of fractional Brownian motion andanomalous diffusion phenomena. Closely related are random walk problemsand the investigation of stochastic processes, where e.g. survival probabili-ties R ( t ) differ from the exponential law, which is known from the classicalPoisson process.c (cid:13) Year Diogenes Co., Sofiapp. xxx–xxx, DOI: .................. R. HerrmannBased on the concept of non-exponential power law Markov-processes,the fractional extension of the classical Poisson process leads to the stan-dard fractional Poisson distribution function of the form [8],[12],[9],[13],[6] p k ( α, λ ) = ( νλ α ) k k ! ∞ X n =0 ( k + n )! n ! ( − νλ α ) n Γ( α ( k + n ) + 1) (1.1)= ( νλ α ) k k ! d k dλ k E α ( λ ) | − νλ α (1.2)= ( − k (cid:0) λ k k ! d k dλ k E α ( λ ) (cid:1) | − νλ α < α ≤ E α ( λ )and its derivatives [10].The mean µ and the variance σ for this fractional Poisson distributionfollow as [6] µ = νλ α Γ(1 + α ) (1.4) σ = µ + µ ( √ π Γ(1 + a )2 α − Γ(1 / α ) −
1) (1.5)The distribution function (1.1) is the result of the underlying processand allows for a direct comparison of experimental data with the corre-sponding model assumptions.In the following, we will consider the inverse problem: We will presentfractional generalizations of distribution functions, based on generalizedMittag-Leffler functions.The motivation for this approach is straight forward: A given exper-imental setup will generate distribution data first, which may be directlycompared with generalized distribution functions. The underlying micro-scopic process may be investigated in a second step.A large amount of literature covers extensions of the standard Mittag-Leffler function by introducing additional parameters and functional be-havior (see e.g. [5], [4] and references therein), but a direct e.g. physicalinterpretation of the additional parameters is a still open question in manycases.In the following we therefore will apply an approach, which allows fora wide class of Mittag-Leffler type functions to determine the correspond-ing distribution functions. This distributions may be directly comparedwith observed phenomena, which will turn out to be helpful for a broaderunderstanding of the functionality and scope of possible applications.ENERALIZED FRACTIONAL POISSON DISTRIBUTION 3
2. Nomenclature
In order to calculate the major properties of a generalized Poisson dis-tribution based on the generalized Mittag-Leffler function, let us first recallthe major properties of the classical Poisson distribution, which is givenby: p k ( λ ) = 1 N λ k k ! (2.1)with the normalization constant N , which is determined by the requirementof normalizability of the distribution, which coincides with the zeroth rawmoment µ : µ = ∞ X k =0 p k ( λ ) = 1 N ∞ X k =0 λ k k ! = 1 N e λ = 1 (2.2)Higher raw moments are then given as: µ n = ∞ X k =0 k n p k ( λ ) = e − λ ∞ X k =0 k n λ k k ! ≡ B n ( λ ) (2.3)where (2.3) is the defining equation for the Bell polynomials B n , whichobey the recursion relation[1] ddx B n ( x ) = 1 x B n +1 ( x ) − B n ( x ) (2.4)and are related to the Stirling numbers of second kind S ( n, k ) [16] viaDobrinski’s formula[2]: B n ( λ ) = n X k =1 S ( n, k ) λ k (2.5)The first raw moments result as: µ = λ (2.6) µ = λ ( λ + 1) (2.7)and therefore the mean µ and variance σ follow as centered moments: µ ≡ µ = λ (2.8) σ ≡ µ − µ = λ ( λ + 1) − λ = λ (2.9)
3. Generalized fractional Poisson distribution
We now extend the definition of the classical Poisson distribution p k ( λ )by applying the canonical fractionalization procedure: R. HerrmannWe replace the faculty in (2.1) by the Γ function, introduce as an ex-ample two additional parameters α, β and define a generalized fractionalPoisson distribution p k ( λ, α, β ) via: p k ( λ, α, β ) = 1 N λ k Γ( β + αk ) (3.1)the normalization constant N follows from the requirement of normaliz-ability of the distribution: µ = ∞ X k =0 p k ( λ, α, β ) = 1 N ∞ X k =0 λ k Γ( β + αk ) = 1 N E α,β ( λ ) = 1 (3.2)with the generalized Mittag-Leffler function E α,β ( λ ) which is defined as[18]: E α,β ( λ ) = ∞ X k =0 λ k Γ( β + αk ) (3.3)The distribution is therefore given by: p k ( λ, α, β ) = 1 E α,β ( λ ) λ k Γ( β + αk ) (3.4)Higher raw moments may be calculated by an iterative procedure, whichwe sketch for the case n = 1.Starting with unnormalized zeroth moment, we apply the operator ddλ λ : E α,β ( λ ) = ∞ X k =0 λ k Γ( β + αk ) (3.5) ddλ λ · E α,β ( λ ) = ∞ X k =0 ( k + 1) λ k Γ( β + αk ) (3.6)Using the commutation relation[ ddλ , λ ] = 1 (3.7)the result for the first raw moment follows as: λ ddλ · E α,β ( λ ) + E α,β ( λ ) = ∞ X k =0 k λ k Γ( β + αk ) + ∞ X k =0 λ k Γ( β + αk ) (3.8) λ ddλ · E α,β ( λ ) = ∞ X k =0 k λ k Γ( β + αk ) (3.9)This procedure may be applied iteratively n times to obtain the n-th rawmoment.ENERALIZED FRACTIONAL POISSON DISTRIBUTION 5We obtain an iterative solution for the raw moments: µ n = 1 E α,β ( λ ) ( λ ddλ ) n · E α,β ( λ ) (3.10)which results in a n-fold application of the Euler-operator J E = x ddx .In order to deduce a closed form solution, we perform a normal orderingof the n-fold Euler operator J nE :With the settings x n → λ n E ( n ) α,β ( λ ) (3.11)= (: λ ddλ :) n E α,β ( λ ) (3.12)where E ( n ) α,β ( λ ) denotes the n-th derivative of the Mittag-Leffler functionwith respect to λ and (::) indicates the normal ordered product [17](: λ ddλ :) n = λ n d n dλ n (3.13)the iteration scheme is now isomorphic to the recurrence relation (2.4) andwe obtain for the raw moments µ n = 1 E α,β ( λ ) B n (: λ ddλ :) · E α,β ( λ ) (3.14)= 1 E α,β ( λ ) k = n X k =1 S ( n, k )( λ n E ( n ) α,β ( λ )) (3.15)The mean µ and variance σ follow as the lowest centered moments: µ ≡ µ = 1 E α,β ( λ ) λE (1) α,β ( λ ) (3.16)= λ ddλ log( E α,β ( λ )) (3.17) σ ≡ µ − µ = 1 E α,β ( λ ) ( λ E (2) α,β ( λ ) + λE (1) α,β ( λ )) − µ (3.18)= ( λ ddλ log( E (1) α,β ( λ )))( λ ddλ log( E α,β ( λ ))) − ( λ ddλ log( E α,β ( λ ))) + λ ddλ log( E α,β ( λ )) (3.19)Optionally, with ddx E α,β ( x ) = 1 α E α,α + β − ( x ) + 1 − βα E α,α + β ( x ) (3.20)all calculations may be reduced to Mittag-Leffler functions directly. R. Herrmann λ = 5β = 1 kpk α varying α>1 α<1 Figure 1.
For λ = 5 and β = 1 the probability distribution p k ( λ = 5 , α, β = 1) from (3.4) is plotted for 1 . ≥ α ≥ . .
05 steps. Thick dashed line indicates α = 1, whichcorresponds to the classical Poisson distribution.
4. Discussion
In contrast to the standard fractional Poisson distribution, which isvalid only in the interval 0 ≤ α ≤
1, our derivation is not restricted to aspecific region of allowed α values. Nevertheless, in the following we presentresults in the practically interesting region 0 < α <
2. In figures 1 and 2 wehave plotted the generalized Poisson distribution (3.4) for varying α and β respectively.To simplify an interpretation, we first calculate the mean and variancefor two limiting cases:First, for small λ ≪ λ → µ = Γ( β )Γ( α + β ) λ (4.1)lim λ → σ = Γ( β )Γ( α + β ) λ (4.2)which shows two interesting features:ENERALIZED FRACTIONAL POISSON DISTRIBUTION 7 λ = 5α = 1 kpk β varying β<1β>1 Figure 2.
For λ = 5 and α = 1 the probability distribution p k ( λ = 5 , α = 1 , β ) from (3.4) is plotted for 4 ≥ β ≥ − . β = 1, which corre-sponds to the classical Poisson distribution.The feature µ = σ , which we know from the classical Poisson distribu-tion, holds for small λ for the generalized case too. For β = 1 the meancorresponds to the mean value, which is obtained with standard fractionalPoisson distribution (1.1), only the shape differs (since variances are differ-ent).However, this case ( λ ≪
1) is a very special case, in general, we dealwith λ ≫ λ ≫
1, using the asymptotic expression for the generalizedMittag-Leffler function[3]lim λ →∞ E α,β ( λ ) = 1 α λ (1 − β ) /α e λ /α + o ( 1 λ ) 0 < α < λ →∞ µ = 1 − β + λ /α α (4.4)lim λ →∞ σ = λ /α α (4.5) R. Herrmann α = 0.1 kpk α = 1.0 standard versusgeneralizedfractional Poissondistribution λ = 5ν = 1 s s Figure 3.
Fit results of the standard fractional Poisson dis-tribution p k ( α s , λ ) (1.1) with ν = 1 for 0 . ≤ α s ≤ . p k ( α, β, λ ) (3.4) (lines) for λ = 5. Fit parameter sets { α.β } are listed in table 1.which allows for a direct interpretation of parameter β : For a given α , theexpectation value µ is shifted by the amount (1 − β ) /α , while the varianceis independent of β . In figure 2, this behavior is easily observed.Consequently, the parameter β could be directly interpreted withinpredator-prey systems: Since the probability distribution is shifted by a β depended constant amount from the expected mean ≈ λ /α /α , whileother characteristics (e.g. shape) of the distribution remain unchanged, β may be considered as a constant time independent population decrease andincrease rate respectively (comparable to a sink/source in electro dynam-ics).With this understanding, in a next step we may then investigate thecorresponding fractional process, considering a fractional pendant of thecoupled sets of Lotka-Volterra equations[11], which are currently used tomodel such scenarios.ENERALIZED FRACTIONAL POISSON DISTRIBUTION 9 Table 1.
Parameter sets for a fit of the standard frac-tional Poisson distribution p k ( α s , λ ) (1.1) with ν = 1 fordifferent α s and the generalized fractional Poisson distribu-tion p k ( α, β, λ ) (3.4) for λ = 5. α s α β α s α β { α, β } in (3.4) may be adjusted appropri-ately to obtain similar mean µ and variance σ values of both, the standardfractional Poisson distribution (1.1) and the generalized fractional Poissondistribution , which leads to similar shapes of the distribution function.This may be realized by calculating the parameter pair { α, β } by equating µ and σ from (1.4), (1.5) with (3.16), (3.18), by the requirement, that bothprobability distributions should be same values for two given k’s or by a fitprocedure, e.g. near the maximum of the distribution.In figure 3 we compare the results of a fitting procedure of the fractionalPoisson distribution (1.1) with (3.4). In table 1 the corresponding fitted pa-rameters are listed. The agreement is very good. In practical applications,it would be difficult, to distinguish both distribution families.Therefore we may conclude, that the proposed generalized Poisson dis-tribution covers the complete parameter range of the standard fractionalPoisson distribution as a subset. It should at least be mentioned, that thenumerical behavior of the proposed generalized fractional Poisson distribu-tion is much easier to handle than the standard distribution. Conclusion
We have presented a general method to derive the characteristic prop-erties of a given fractional distribution in terms of normal ordered Bellpolynomials with the Euler-operator as an argument.We have demonstrated for the special case of the generalized Mittag-Leffler function, that the investigation of corresponding fractional distri-butions helps to obtain a clearer understanding of first the meaning ofadditional parameters in generalizations of Mittag-Leffler type functionsand second of possible areas of their application.0 R. HerrmannIn addition we have shown, that the standard fractional Poisson distri-bution may be reproduced as a specific subset of the generalized fractionalPoisson distribution.
Acknowledgements
We thank A. Friedrich for valuable discussions.
References [1] E. T. Bell, (1934) Exponential polynomials
Ann. Math. (2) 258–277.[2] G. Dobinski, (1877) Summirung der Reihe P n m /n ! f¨ur m =1 , , , , , ... Grunert Archiv (Arch. Math. Phys.) arXiv:0909.0230, Journal of AppliedMathemathics, Hindawi , (2011) 298628.[4] A. A. Kilbas, A. A. Koroleva and S. S. Rogosin, (2013) Multi-parameterMittag-Leffler functions and their extension Fract. Calc. Appl. Anal. Fract. Calc.Appl. Anal. Commun. Nonlin. Sci.Num. Sim J. Math. Phys. Phys. Rev. E (1999) 5026.[9] R. Metzler and J. Klafter, The random walk’s guide to anomalous dif-fusion: a fractional dynamics approach. Phys. Rep. (2000) 1–77;DOI:10.1016/S0370-1573(00)00070-3[10] M. G. Mittag-Leffler, (1903) Sur la nouvelle function E α ( x ) ComptesRendus Acad. Sci. Paris
Fractional Differential Equations
Academic Press,Boston[13] O. N. Repin and A. I. Saichev, (2000) Fractional Poisson law
Radio-phys. Quant. Electron. ”The Exponential Polynomials” and ”The Bell Poly-nomials.” 4.1.3 and 4.1.8 in The Umbral Calculus. New York: Aca-demic Press, pp. 63–67 and 82–87.ENERALIZED FRACTIONAL POISSON DISTRIBUTION 11[15] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integralsand derivatives
Translated from the 1987 Russian original, Gordon andBreach, Yverdon(1993)[16] J. Stirling, (1730) Methodus differentialis, sive tractatus de summationet interpolation serierum infinitarium. London. English translation byHolliday, J. The Differential Method: A Treatise of the Summation andInterpolation of Infinite Series. 1749.[17] G. C. Wick, (1950) The evaluation of the collision matrix
Phys. Rev. E a ( x ) Acta Math.1