Information Technology Reference
In-Depth Information
=(0
m
0
1
m
1
...n
m
n
):
We now associate several coecients with
λ
=
{{
λ
1
,...,λ
k
}}
A
(
λ
)=
n
λ
1
,...,λ
k
B
(
λ
)=
k
m
0
,m
1
,...,m
n
C
(
λ
)=
A
(
λ
)
B
(
λ
)
m
0
!
k
!
D
(
λ
)=
k
!
m
0
!
Now let us define the
L
(loosen) and
R
(roughen) operators:
L
:(
x
1
,...,x
k
)
→{{
x
1
,...,x
k
}}
R
:(
y
1
,...,y
k
)
→
(
|
y
1
|
,...,
|
y
k
|
)
Theorem.
As summarized in the diagram below,
(a) For all
c
C
∗
,
R
−
1
(
c
)
|
|
=
A
(
c
).
∈
P
∗
,
L
−
1
(
λ
)
(b) For all
λ
|
|
=
B
(
λ
).
∈
P
∗
,
R
−
1
(
λ
)
(c) For all
λ
|
|
=
C
(
λ
).
∈
∗
,
∈
P
L
−
1
(
π
)
(d) For all
π
|
|
=
D
(
π
).
L
C
∗
−→
P
∗
|
↓
|
↓
R
R
L
C
∗
P
∗
−→
Corollaries
f
[
π
]=
λ∈P
n
C
(
λ
)
f
[
λ
]
π∈
P
n
f
[
α
]=
c
A
(
c
)
f
[
c
]=
λ
A
(
λ
)
B
(
λ
)
f
[
λ
]
α
∈
C
n
∈
C
n
∈
P
n
.
6
Illustrative Example
In this section we will illustrate in some detail one of the entries given in Table 2.
Specifically, given a Taylor series
F
(
z
) and assuming
F
(0) = 1, we seek the Taylor
coecients for
G
(
z
)=
F
(
z
)
−
2
, i.e.,
g
[
n
]:=
x
n
n
!
F
(
z
)
−
2
,
for
n
≥
0.
According to Table 2A,
−
f
[
g
[
n
]=
α∈
C
n
2
|
α
1
|
]
···
f
[
|
α
k
|
]
,
(4)
|
α
|
