Page 174 - Determinants and Their Applications in Mathematical Physics

4.12 Hankelians 5 159

D {Q n (t, t)} = C n (t) C 1 (t) C 2 (t) ··· C n−1 (t)
n

τ
n

=(−1) n−1 C 1 (t) C 2 (t) ··· C n−1 (t) C n (t)
n
=(−1) n−1 F n . (4.12.23)
(n)
The cofactors Q ,1 ≤ i ≤ n, are independent of τ.
i1
(n) (n)
Q (t)= E = G n−1 ,
11 11
(n)
Q n1 (t)=(−1) n+1 C 1 (t) C 2 (t) C 3 (t) ··· C n−1 (t) n−1
=(−1) n+1 F n−1 ,
(n)
Q (t, τ)=(−1) n+1 C 1 (τ) C 2 (t) C 3 (t) ··· C n−1 (t) .(4.12.24)
n−1
1n
Hence,
(n)
D {Q (t, t)} =0, 1 ≤ r ≤ n − 2
r
τ
1n
D n−1 {Q (n) (t, t)} =(−1) n+1 C n (t) C 2 (t) C 3 (t) ··· C n−1 (t)
n−1
τ 1n

= − C 2 (t) C 3 (t) ··· C n−1 (t) C n (t)

n−1
= −G n−1 ,
(n)
D {Q (t, t)} = −D t (G n−1 ),
n
τ 1n
Q (n) (t, τ)= Q n−1 (t, τ),
nn
Q (n) (t, t)= E n−1 ,
nn
0, 1 ≤ r ≤ n − 2

D {Q (n) (t, t)} = (−1) F n−1 , r = n − 1 (4.12.25)
r
n
τ nn
(−1) D t (F n−1 ),r = n.
n
(n)
Q (t)= G n−2 . (4.12.26)
1n,1n
Applying the Jacobi identity to the cofactors of the corner elements of
Q n ,
(n) (n)
Q 11 (t) Q (t, τ) (n)
1n = Q n (t, τ)Q (t),
(n) (n)
Q n1 (t) Q nn (t, τ)
1n,1n
(n)
G n−1
Q (t, τ)
1n
(n) = Q n (t, τ)G n−2 . (4.12.27)
(−1) F n−1 Q nn (t, τ)
n+1
The first column of the determinant is independent of τ, hence, differenti-
ating n times with respect to τ and putting τ = t,

G n−1 n+1
D t (G n−1 )
n+1 =(−1) F n G n−2 ,
n
(−1) F n−1 (−1) D t (F n−1 )
G n−1 D t (F n−1 ) − F n−1 D t (G n−1 )= −F n G n−2 ,

G n−1 F n G n−2
= 2 .
D t
F n−1 F
n−1