for the stable law g
ν
·; y defined above, we get f
ν k
t = Z
+∞
g
ν
t; y
λ y
k −1
e
− y
k − 1 d y.
2.25 Formula 2.25 permits us to conclude that f
ν k
t can be interpreted as the law of the stable random variable S
ν
with a random scale parameter possessing an Erlang distribution.
2.3 The probability generating function
We consider now the equation governing the probability generating function, defined, for any |u| ≤
1, as G
ν
u, t =
∞
X
k=0
u
k
p
ν k
t. 2.26
From 2.1 it is straightforward that it coincides with the solution to the fractional differential equation
∂
ν
Gu, t ∂ t
ν
= λu − 1Gu, t, ν ≤ 1
2.27 subject to the initial condition Gu, 0 = 1. As already proved in [2] the Laplace transform of G
ν
= G
ν
u, t is given by L
G
ν
u, t; s =
s
ν−1
s
ν
− λu − 1 2.28
so that the probability generating function can be expressed as G
ν
u, t = E
ν,1
λu − 1t
ν
, |u| ≤ 1, t 0.
2.29 By considering 2.29 together with the previous results we get the following relationship, valid for
the infinite sum of GML functions:
∞
X
k=0
λut
ν k
E
k+1 ν,ν k+1
−λt
ν
= E
ν,1
λu − 1t
ν
. 2.30
Formula 2.30 suggests a useful general relationship between the infinite sum of GML functions and the standard Mittag-Leffler function:
∞
X
k=0
ux
k
E
k+1 ν,ν k+1
−x = E
ν,1
xu − 1, |u| ≤ 1, x
0. 2.31
By considering the derivatives of the probability generating function 2.29 we can easily derive the factorial moments of
N
ν
which read E
N
ν
tN
ν
t − 1...N
ν
t − r + 1 =
λt
ν r
r Γν r + 1
. 2.32
These are particularly useful in checking that N
ν
represents a Cox process with directing measure Λ. Indeed, as pointed out in [12], the factorial moments of a Cox process coincide with the ordinary
692
moments of its directing measure. We show that this holds for N
ν
, by using the contour integral representation of the inverse of Gamma function,
E [Λ0, t]]
r
= Z
+∞
y
r
v
2 ν
y, td y =
Z
+∞
y
r
λt
ν
W
−ν,1−ν
−
y λt
ν
d y
= 1
λt
ν
1 2
πi Z
+∞
y
r
d y Z
H a
e
z −
y t−ν λ
z
ν
z
1 −ν
dz =
λ
r
2 πi
Z
H a
e
z
z
1+ ν r
dz Z
+∞
t
ν r
w
r
e
−w
d w =
λ
r
t
ν r
2 πi
Γr + 1 Z
H a
e
z
z
1+ ν r
dz = λt
ν r
r Γν r + 1
, which coincides with 2.32.
For r = 1 we can obtain from 2.32 the renewal function of the process N
ν
, which reads m
ν
t = EN
ν
t=−λt
ν
d d
µ E
ν,1
λe
−µ
− 1t
ν µ=0
2.33 =
λt
ν
d d x
E
ν,1
x
x=0
= λt
ν
Γν + 1 ,
and coincides with that obtained in [18], for λ = 1. It is evident also from 2.33 that the mean
waiting time which is equal to lim
t →∞
t m
ν
t is infinite, since ν 1.
3 Second-type fractional recursive differential equation
3.1 The solution
