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Asymptotic Behavior
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Asymptotic Behavior
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Asymptotic expansion for a linear first-order ordinary difference equation (O
Δ
E) with an irregular singular point at infinity:
I
n
[
1
]
:
=
s
=
A
s
y
m
p
t
o
t
i
c
R
S
o
l
v
e
V
a
l
u
e
[
y
[
n
+
1
]
3
n
y
[
n
]
,
y
[
n
]
,
{
n
,
∞
,
4
}
]
O
u
t
[
1
]
=
n
3
1
-
5
7
1
2
4
8
8
3
2
0
4
n
-
1
3
9
5
1
8
4
0
3
n
+
1
2
8
8
2
n
+
1
1
2
n
-
1
2
+
n
n
1
Plot the solution using a logarithmic scale:
I
n
[
2
]
:
=
D
i
s
c
r
e
t
e
P
l
o
t
C
a
l
l
o
u
t
s
/
.
{
1
1
}
,
T
r
a
d
i
t
i
o
n
a
l
F
o
r
m
[
]
,
{
n
,
2
^
R
a
n
g
e
[
1
,
9
]
}
,
S
c
a
l
i
n
g
F
u
n
c
t
i
o
n
s
{
"
L
o
g
"
,
"
L
o
g
"
}
,
O
u
t
[
2
]
=
Plot the solution of an asymptotic expansion for a linear third-order O
Δ
E with an irregular singular point:
I
n
[
3
]
:
=
s
=
A
s
y
m
p
t
o
t
i
c
R
S
o
l
v
e
V
a
l
u
e
[
y
[
n
+
3
]
-
y
[
n
]
-
(
n
+
2
)
y
[
n
+
1
]
0
,
y
[
n
]
,
{
n
,
∞
,
1
}
]
O
u
t
[
3
]
=
n
(
-
)
1
-
1
3
1
2
n
-
3
2
-
n
n
1
+
-
n
/
2
1
+
5
1
2
n
-
1
n
n
/
2
n
2
+
n
(
-
1
)
-
n
/
2
1
+
5
1
2
n
+
1
n
n
/
2
n
3
I
n
[
4
]
:
=
D
i
s
c
r
e
t
e
P
l
o
t
C
a
l
l
o
u
t
N
[
s
/
.
{
1
1
,
2
1
,
3
1
}
,
5
]
,
T
r
a
d
i
t
i
o
n
a
l
F
o
r
m
[
]
,
{
n
,
2
^
R
a
n
g
e
[
9
]
}
,
S
c
a
l
i
n
g
F
u
n
c
t
i
o
n
s
{
"
L
o
g
"
,
"
L
o
g
"
}
,
O
u
t
[
4
]
=
"
"
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