Fractional negative binomial and pólya processes
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...th some special functions that will be required later. The Mittag-Leffler function Lβ(z) is defined as (see [8]) (2.1) Lβ(z) = X∞ k=0 zk Γ(1+βk) , β,z∈ Cand Re(β) >0. The M-Wright function Mβ(z) (see [10, 17]) is defined as Mβ(z) = X∞ n=0 (−z)n n!Γ(−βn+(1− β)) = 1 π X∞ n=1 (−z)n−1 (n− 1)! Γ(βn)sin(πβn), z∈ C, 0 <β<1. Fractional Negative Binomial and Polya Processes 3 Let p,q∈ Z+\{0}. Also, for 1 ≤ i≤...
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"Fractional negative binomial and pó..." refers background in this paper
... Wright function defined in (2.5). It is also known that (see [16]) Nβ(t,λ) = N(Eβ(t),λ), where Eβ(t) is the hitting time of a stable subordinator Dβ(t). The mean and variance of FPP are given by (see [12]) ENβ(t,λ) = λtβ Γ(β+1) (3.5) ; Var(Nβ(t,λ)) = λtβ Γ(β+1) ˆ 1+ λtβ Γ(β+1) βB(β,1/2) 22β−1 − 1 ˙ (3.6) , where B(a,b) denotes the beta function. First we establish an important property of an FPP. 3...
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...gative binomial process, with Q(t) ∼ NB(t|η,pt), where η= α/(1+ α). Let 0 <β<1. A natural generalization is to consider Qβ(t) = Nβ(Γ(t),λ), where {Nβ(t)}t≥0 is a fractional Poisson process (see [12, 16]), and we call {Qβ(t)}t≥0 is a fractional negative binomial process (FNBP). We will show that this process is different from the FNBP defined in [3] and [4]. It is known that the Polya process is obtain...
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...t 0 <β<1. The fractional Poisson process (FPP) {Nβ(t,λ)}t≥0, which is a generalization of the Poisson process {N(t,λ)}t≥0 and solves the following fractional difference-differential equation (see [12, 14, 16]) Dβ t p β (n|t,λ) = −λp β (n|t,λ)+λp β (3.1) (n− 1|t,λ), for n≥ 1 Dβ t p β (0|t,λ) = −λp β (0|t,λ), where p β (n|t,λ) = P{Nβ(t,λ) = n} and (3.2) Dβ t u(t,x) = 1 Γ(1− β) Z t 0 ∂u(r,x) ∂r dr (t− r)β , ...
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