Page 199 - Determinants and Their Applications in Mathematical Physics

184 5. Further Determinant Theory

n+1−j

ψ 2n−s δ s,n−j+1
= H n
s=1
= H n ψ n+j−1 , 1 ≤ j ≤ n; (5.2.21)
2n−i−j+1
n s = t − j
(n)
Y i = K φ t+i+2 ψ 2n−s ,
s → t
jn
j=1 s=n+2−j
2n−i+1
n
(n)
= K
jn φ t+i−2 ψ 2n+j−t
j=1 t=n+2
2n−i+1 n
(n)

= φ t+i−2 ψ 2n+j−t K
jn
t=n+2 j=1
2n−i+1 n
(n)
= φ t+i−2 k j+n+1−t,n K
jn
t=n+2 j=1
2n−i+1

φ t+i−2 δ t,n+1
= K n
t=n+2
=0, 1 ≤ i ≤ n − 1, (5.2.22)
since t>n + 1. Hence,
n
(n)
K ψ n+j−1
Pf n = H n
jn
j=1
n
(n)
k jn K
= H n
jn
j=1
= H n K n ,
which completes the proof of Theorem 5.1.
5.2.4 Three Further Theorems
The principal theorem, when expressed in the form
2n−1
i+1 (n)
(−1) Pf a i,2n = H n K n , (5.2.23)
i
i=1
yields two corollaries by partial differentiation. Since the only elements
in Pf n which contain φ 2n−1 and ψ 2n−1 are a i,2n ,1 ≤ i ≤ 2n − 1, and
(n) (n)
Pf is independent of a i,2n , it follows that Pf is independent of φ 2n−1
i i
and ψ 2n−1 . Moreover, these two functions occur only once in H n and K n ,
respectively, both in position (n, n).