4.12 Hankelians 5 159
D
n
τ
{Qn(t, t)}=
∣
∣C
n(t)C 1 (t)C 2 (t)···Cn− 1 (t)
∣
∣
n
=(−1)
n− 1
∣
∣C
1 (t)C 2 (t)···Cn− 1 (t)Cn(t)
∣
∣
n
=(−1)
n− 1
Fn. (4.12.23)
The cofactorsQ
(n)
i 1
,1≤i≤n, are independent ofτ.
Q
(n)
11
(t)=E
(n)
11
=Gn− 1 ,
Q
(n)
n 1
(t)=(−1)
n+1
∣
∣C
1 (t)C 2 (t)C 3 (t)···Cn− 1 (t)
∣
∣
n− 1
=(−1)
n+1
Fn− 1 ,
Q
(n)
1 n
(t, τ)=(−1)
n+1
∣
∣C
1 (τ)C 2 (t)C 3 (t)···Cn− 1 (t)
∣
∣
n− 1
.(4.12.24)
Hence,
D
r
τ{Q
(n)
1 n(t, t)}=0,^1 ≤r≤n−^2
D
n− 1
τ
{Q
(n)
1 n
(t, t)}=(−1)
n+1
∣
∣C
n(t)C 2 (t)C 3 (t)···Cn− 1 (t)
∣
∣
n− 1
=−
∣
∣
C 2 (t)C 3 (t)···Cn− 1 (t)Cn(t)
∣
∣
n− 1
=−Gn− 1 ,
D
n
τ{Q
(n)
1 n
(t, t)}=−Dt(Gn− 1 ),
Q
(n)
nn
(t, τ)=Qn− 1 (t, τ),
Q
(n)
nn(t, t)=En−^1 ,
D
r
τ
{Q
(n)
nn
(t, t)}=
{
0 , 1 ≤r≤n− 2
(−1)
n
Fn− 1 ,r=n− 1
(−1)
n
Dt(Fn− 1 ),r=n.
(4.12.25)
Q
(n)
1 n, 1 n
(t)=Gn− 2. (4.12.26)
Applying the Jacobi identity to the cofactors of the corner elements of
Qn,
∣
∣
∣
∣
Q
(n)
11
(t) Q
(n)
1 n
(t, τ)
Q
(n)
n 1
(t) Q
(n)
nn(t, τ)
∣
∣
∣
∣
=Qn(t, τ)Q
(n)
1 n, 1 n
(t),
∣
∣
∣
∣
Gn− 1 Q
(n)
1 n
(t, τ)
(−1)
n+1
Fn− 1 Q
(n)
nn(t, τ)
∣
∣
∣
∣
=Qn(t, τ)Gn− 2. (4.12.27)
The first column of the determinant is independent ofτ, hence, differenti-
atingntimes with respect toτand puttingτ=t,
∣
∣
∣
∣
Gn− 1 Dt(Gn− 1 )
(−1)
n+1
Fn− 1 (−1)
n
Dt(Fn− 1 )
∣
∣
∣
∣
=(−1)
n+1
FnGn− 2 ,
Gn− 1 Dt(Fn− 1 )−Fn− 1 Dt(Gn− 1 )=−FnGn− 2 ,
Dt
[
Gn− 1
Fn− 1
]
=
FnGn− 2
F
2
n− 1