4.9 Hankelians 2 121
Proof. The sum formula forTcan be expressed in the form
n
∑
j=1
ψr+i+j− 1 T
(n,r)
ij
=−δinT
(n+1,r)
n+1,n
, (4.9.10)
C
′
j=
[
(r+j)ψr+j(r+j+1)ψr+j+1···(r+j+n−1)ψr+j+n− 1
]T
n
.(4.9.11)
Let
C
∗
j=C
′
j−(r+j)Cj+1
=
[
0 ψr+j+1 2 ψr+j+2···(n−1)ψr+j+n− 1
]T
n
. (4.9.12)
Differentiating the columns ofT,
T
′
=
n
∑
j=1
Uj,
where
Uj=
∣
∣C
1 C 2 ···C
′
j
Cj+1···Cn
∣
∣
n
, 1 ≤j≤n.
Let
Vj=
∣
∣C
1 C 2 ···C
∗
j
Cj+1···Cn
∣
∣
n
, 1 ≤j≤n
=
n
∑
i=2
(i−1)ψr+i+j− 1 Tij. (4.9.13)
Then, performing an elementary column operation onUj,
Uj=Vj, 1 ≤j≤n− 1
Un=
∣
∣C
1 C 2 ···Cn− 1 C
′
n
∣
∣
=
∣
∣C
1 C 2 ···Cn− 1 C
∗
n
∣
∣+(r+n)
∣
∣C
1 C 2 ···Cn− 1 Cn+1
∣
∣
=Vn−(r+n)T
(n+1,r)
n+1,n
. (4.9.14)
Hence,
T
′
+(r+n)T
(n+1,r)
n+1,n
=
n
∑
j=1
Vj
=
n
∑
j=1
(i−1)
n
∑
j=1
ψr+i+j− 1 Tij
=−T
(n+1,r)
n+1,n
n
∑
i=2
(i−1)δin
=−(n−1)T
(n+1,r)
n+1,n.
The theorem follows.