Page 136 - Determinants and Their Applications in Mathematical Physics

4.9 Hankelians 2 121

Proof. The sum formula for T can be expressed in the form
n
(n,r) (n+1,r)
ψ r+i+j−1 T = −δ in T , (4.9.10)
ij n+1,n
j=1

T
C = (r+j)ψ r+j (r+j +1)ψ r+j+1 ··· (r+j +n−1)ψ r+j+n−1 . (4.9.11)
j
n
Let
∗
C = C − (r + j)C j+1
j j

T
= 0 ψ r+j+1 2ψ r+j+2 ··· (n − 1)ψ r+j+n−1 . (4.9.12)
n
Differentiating the columns of T,
n

T = U j ,
j=1
where

U j = C 1 C 2 ··· C C j+1 ··· C n , 1 ≤ j ≤ n.

j
n
Let

V j = C 1 C 2 ··· C C j+1 ··· C n , 1 ≤ j ≤ n
∗

j
n
n

= (i − 1)ψ r+i+j−1 T ij . (4.9.13)
i=2
Then, performing an elementary column operation on U j ,
U j = V j , 1 ≤ j ≤ n − 1

U n = C 1 C 2 ··· C n−1 C

n

= C 1 C 2 ··· C n−1 C ∗ +(r + n) C 1 C 2 ··· C n−1 C n+1

n
(n+1,r)
= V n − (r + n)T . (4.9.14)
n+1,n
Hence,
n

T +(r + n)T (n+1,r) =
n+1,n V j
j=1
n n

= (i − 1) ψ r+i+j−1 T ij
j=1 j=1
n
(n+1,r)
= −T
n+1,n (i − 1)δ in
i=2
(n+1,r)
= −(n − 1)T .
n+1,n
The theorem follows.