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)
α
in terms of the
Equation (
4.45
) is made usable by interpreting the operator
(
1
−
B
binomial theorem when the power-law index is an integer,
1
2
1
3
)
α
=
B
2
B
3
(
1
−
B
1
−
α
B
−
!
α(
1
−
α)
−
!
α(
1
−
α)(
2
−
α)
−···
n
∞
n
=
(
−
B
)
,
(4.46)
n
=
0
where, of course, this procedure is defined only when operating on an appropriate
function. The binomial coefficient for a positive integer
α
is formally given by
n
(α
+
1
)
=
)
,
(4.47)
(
n
+
1
)(α
+
1
−
n
where
denotes a gamma function. Note that for negative integers the gamma
function has poles so that the binomial coefficient is zero if
n
(
·
)
>α
in (
4.46
) and
α
is integer-valued. In the case of arbitrary complex
α
and
β
with the restriction
α
=−
1
,
−
2
,...,
we can write
α
β
(α
+
1
)
sin
[
(β
−
α)π
]
π
(α
+
)(β
−
α)
(β
+
1
=
−
β)
=
,
(4.48)
(β
+
)(α
+
)
1
1
1
which is readily established using the Euler integral of the second kind
∞
x
z
−
1
e
−
x
dx
(
z
)
=
;
Re
z
>
0.
(4.49)
0
Using this extension to the complex domain for the gamma function, the formal expres-
sion (
4.46
) is equally valid; however, the series no longer cuts off for
n
>α
and is an
infinite series for non-integer values of the power-law index.
The fractional difference equation has the formal solution
)
−
α
ξ
j
,
Q
n
+
1
=
(
−
1
B
(4.50)
where we define the inverse operator
∞
∞
0
θ
k
B
k
Q
j
+
1
=
ξ
j
=
0
θ
k
ξ
j
−
k
.
(4.51)
k
=
k
=
Here we see that the solution to the fractional difference equation at time
j
1is
determined by random fluctuations that extend infinitely far back into the past with the
strength of that influence being determined by the functional dependence of the coeffi-
cients
+
θ
k
on
k
. These coefficients are determined from the convergent expansion of the
function
)
−
α
for
θ(
z
)
=
(
1
−
z
|
α
|
<
1
/
2,
−
n
∞
)
−
α
=
n
(
1
−
z
(
−
z
)
,
(4.52)
n
=
0
and using some identities among gamma functions,
−
n
n
(
1
−
α)
)
=
(
−
1
)
(α
+
n
)
=
)(α)
,
(
n
+
1
)(
1
−
α
−
n
(
n
+
1
