An interesting relationship between the two models analyzed here can be established by observing that the waiting-time of the k-th event of the process governed by 1.7 coincides in distribution
with the waiting time of the 2k-th event for the first model. This suggests to interpret c N
ν
as a fractional Poisson process of the first type, which jumps upward at even-order events A
2k
and the probability of the successive odd-indexed events A
2k+1
is added to that of A
2k
. As a consequence, the distribution of c
N
ν
can be expressed, in terms of the processes N and T
2 ν
, as follows: Pr
¦ c
N
ν
t = k ©
= Pr N
T
2 ν
t = 2k + Pr
N T
2 ν
t = 2k + 1 ,
k ≥ 0.
We also study the probability generating functions of the two models, which are themselves solu- tions to fractional equations; in particular in the second case an interesting link with the fractional
telegraph-type equation is explored.
For ν = 1, equation 1.7 takes the following form
d
2
p
k
d t
2
+ 2λ d p
k
d t = −λ
2
p
k
− p
k −1
, k
≥ 0 and the related process can be regarded as a standard Poisson process with Gamma-distributed
interarrival times with parameters λ, 2. This is tantamount to attributing the probability of odd-
order values A
2k+1
of a standard Poisson process to the events labelled by 2k. Moreover, it should be stressed that, in this special case, the equation satisfied by the probability generating function
b Gu, t, t
0, |u| ≤ 1, i.e. ∂
2
Gu, t ∂ t
2
+ 2λ ∂ Gu, t
∂ t = λ
2
u − 1Gu, t, ν ≤ 1
coincides with that of the damped oscillations. All the previous results are further generalized to the case n
2 in the concluding remarks: the structure of the process governed by the equation
d
n ν
p
k
d t
n ν
+ n
1 λ
d
n−1ν
p
k
d t
n−1ν
+ ... + n
n − 1
λ
n −1
d
ν
p
k
d t
ν
= −λ
n
p
k
− p
k −1
, k
≥ 0, 1.8
where ν ∈ 0, 1], is exactly the same as before and all the previous considerations can be easily
extended.
2 First-type fractional recursive differential equation
2.1 The solution
We begin by considering the following fractional recursive differential equation d
ν
p
k
d t
ν
= −λp
k
− p
k −1
, k
≥ 0, 2.1
with p
−1
t = 0, subject to the initial conditions p
k
0 = ¨
1 k = 0
k ≥ 1
. 2.2
687
We apply in 2.1 the definition of the fractional derivative in the sense of Caputo, that is, for m ∈ N,
d
ν
d t
ν
ut =
1 Γm−ν
R
t 1
t−s
1+ ν−m
d
m
ds
m
usds, for m
− 1 ν m
d
m
d t
m
ut, for
ν = m .
2.3 We note that, for
ν = 1, 2.1 coincides with the equation governing the homogeneous Poisson process with intensity
λ 0. We will obtain the solution to 2.1-2.2 in terms of GML functions defined in 1.6 and show
that it represents a true probability distribution of a process, which we will denote by N
ν
t, t 0 : therefore we will write
p
ν k
t = Pr N
ν
t = k ,
k ≥ 0, t 0.
2.4
Theorem 2.1 The solution p
ν k
t, for k = 0, 1, ... and t ≥ 0, of the Cauchy problem 2.1-2.2 is given by
p
ν k
t = λt
ν k
E
k+1 ν,ν k+1
−λt
ν
, k
≥ 0, t 0. 2.5
Proof By taking the Laplace transform of equation 2.1 together with the condition 2.2, we obtain
L ¦
p
ν k
t; s ©
= Z
∞
e
−st
p
ν k
td t = λ
k
s
ν−1
s
ν
+ λ
k+1
2.6 which can be inverted by using formula 2.5 of [24], i.e.
L n
t
γ−1
E
δ β,γ
ωt
β
; s o
= s
βδ−γ
s
β
− ω
δ
, 2.7
where Re β 0, Reγ 0, Reδ 0 and s |ω|
1 Re
β
for β = ν, δ = k + 1 and γ = ν k + 1. Therefore the inverse of 2.6 coincides with 2.5.
Remark 2.1 For any ν ∈ 0, 1], it can be easily seen that result 2.5 coincides with formula 2.10 of
[2], which was obtained by a different approach. Moreover Theorem 2.1 shows that the first model proposed by Mainardi et al. [17] as a fractional ver-
sion of the Poisson process called renewal process of Mittag-Leffler type has a probability distribution coinciding with the solution of equation 2.1 and therefore with 2.5.
We derive now an interesting relationship between the GML function in 2.5 and the Wright func- tion
W
α,β
x =
∞
X
k=0
x
k
kΓ αk + β
, α −1, β 0, x ∈ R.
Let us denote by v
2 ν
= v
2 ν
y, t the solution to the Cauchy problem
∂
2 ν
v ∂ t
2 ν
= λ
2 ∂
2
v ∂ y
2
, t
0, y ∈ R v y, 0 =
δ y, for 0
ν 1 v
t
y, 0 = 0, for 1
2 ν 1 .
2.8
688
then it is well-known see [14] and [15] that the solution of 2.8 can be written as v
2 ν
y, t = 1
2 λt
ν
W
−ν,1−ν
− | y|
λt
ν
, t
0, y ∈ R. 2.9
In [2] the following subordinating relation has been proved: p
ν k
t = Z
+∞
e
− y
y
k
k v
2 ν
y, td y = Pr N
T
2 ν
t = k ,
k ≥ 0,
2.10 where
v
2 ν
y, t = ¨
2v
2 ν
y, t, y
0, y
2.11 is the folded solution of equation 2.8. In 2.10
T
2 ν
t, t 0 represents a random time indepen- dent from the Poisson process N with transition density given in 2.9 and 2.11. This density can
be alternatively expressed in terms of the law g
ν
·; y of a a stable random variable S
ν
µ, β, σ of order
ν, with parameters µ = 0, β = 1 and σ =
y λ
cos
πν 2
1 ν
,as v
2 ν
y, t = 1
Γ1 − ν Z
t
t − w
−ν
g
ν
w; y
λ d w
2.12 see [20], formula 3.5, for details. By combining 2.5 and 2.10, we extract the following
integral representation of the GML functions, in terms of Wright functions:
E
k+1 ν,ν k+1
−λt
ν
= 1
k λ
k+1
t
νk+1
Z
+∞
e
− y
y
k
W
−ν,1−ν
− y
λt
ν
d y. 2.13
Remark 2.2 Since result 2.13 holds for any t 0, we can choose t = 1, so that we get, by means of
a change of variable, E
k+1 ν,ν k+1
−λ = 1
k Z
+∞
e
−λ y
y
k
W
−ν,1−ν
− yd y. This shows that the GML function E
k+1 ν,ν k+1
can be interpreted as the Laplace transform of the function
y
k
k
W
−ν,1−ν
− y. In particular, for ν =
1 2
, since 2.10 reduces to Pr
¦ N
1 2
t = k ©
= Z
+∞
e
− y
y
k
k e
− y
2
4λ
2
t
p πλ
2
t d y = Pr
N |B
λ
t| = k ,
where B
λ
t is a Brownian motion with variance 2λ
2
t independent of N , we get for t = 1 E
k+1
1 2
,
k 2
+1
−λ = 1
k Z
+∞
e
−λ y
y
k
e
− y
2
4
p π
d y. 2.14
The previous relation can be checked directly, by performing the integral in 2.14.
689
2.2 Properties of the corresponding process
