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Poisson-type processes governed by fractional and higher-order recursive differential equations. (English) Zbl 1228.60093
Summary: We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e.,
$\frac {d}{dt}p_k(t)= -\lambda \big(p_k(t)-p_{k-1}(t)\big), \quad k\geq 0,\;t>0,$ by introducing fractional time-derivatives of order $$\nu,2\nu,\dots,n\nu$$. We show that the so-called “generalized Mittag-Leffler functions” $$E_{\alpha,\beta}^k(x)$$, $$x\in\mathbb R$$ (introduced by T. R. Prabhakar [Yokohama Math. J. 19, 7–15 (1971; Zbl 0221.45003)]) arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the interarrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for $$t\rightarrow\infty$$. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter $$\nu$$ varying in $$(0,1]$$. For integer values of $$\nu$$, these models can be viewed as a higher-order Poisson processes connected with the standard case by simple and explicit relationships.

##### MSC:
 60K05 Renewal theory 33E12 Mittag-Leffler functions and generalizations 26A33 Fractional derivatives and integrals
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