4.11 Hankelians 4 145
where
C
′
n=λn−^1 ,n−^1 Cn+
n− 1
∑
j=1
λn− 1 ,j− 1 Cj
=
n− 1
∑
j=1
λn− 1 ,j− 1
[
νj− 1 νj···νn+j− 3 νn+j− 2
]T
n
=2
−(4n−5)
[
00 ··· 01
]T
n
. (4.11.42)
Hence,
An=2
−4(n−1)+1
An− 1
An− 1 =2
−4(n−2)+1
An− 2
.......................................
A 2 =2
−4(1)+1
A 1 , (A 1 =ν 0 =1).
(4.11.43)
The theorem follows by equating the product of the left-hand sides to the
product of the right-hand sides.
It is now required to evaluate the cofactors ofAn.
Theorem 4.49.
a.A
(n)
nj
=2
−(n−1)(2n−3)
λn− 1 ,j− 1 ,
b.A
(n)
n 1 =2
−(n−1)(2n−3)
,
c. A
nj
n =2
2(n−1)
λn− 1 ,j− 1.
Proof. Thenequations in (4.11.40) can be expressed in matrix form as
follows:
AnLn=C
′
n, (4.11.44)
where
Ln=
[
λn 0 λn 1 ···λn,n− 2 λn,n− 1
]
T
n
. (4.11.45)
Hence,
Ln=A
− 1
nC
′
n
=A
− 1
n
[
A
(n)
ji
]
n
C
′
n
=2
(n−1)(2n−1)−2(n−1)
[
An 1 An 2 ···An,n− 1 Ann
]T
n
, (4.11.46)
which yields part (a) of the theorem. Parts (b) and (c) then follow
easily.
Theorem 4.50.
A
(n)
ij =2
−n(2n−3)
[
2
2 i− 3
λi− 1 ,j− 1 +
n− 1
∑
r=i+1
λr− 1 ,i− 1 λr− 1 ,j− 1
]
,j≤i<n− 1.