Determinants and Their Applications in Mathematical Physics

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

.