184 5. Further Determinant Theory
=Hn
n+1−j
∑
s=1
ψ 2 n−sδs,n−j+1
=Hnψn+j− 1 , 1 ≤j≤n; (5.2.21)
Yi=
n
∑
j=1
K
(n)
jn
2 n−i−j+1
∑
s=n+2−j
φt+i+2ψ 2 n−s,
[
s=t−j
s→t
]
=
n
∑
j=1
K
(n)
jn
2 n−i+1
∑
t=n+2
φt+i− 2 ψ 2 n+j−t
=
2 n−i+1
∑
t=n+2
φt+i− 2
n
∑
j=1
ψ 2 n+j−tK
(n)
jn
=
2 n−i+1
∑
t=n+2
φt+i− 2
n
∑
j=1
kj+n+1−t,nK
(n)
jn
=Kn
2 n−i+1
∑
t=n+2
φt+i− 2 δt,n+1
=0, 1 ≤i≤n− 1 , (5.2.22)
sincet>n+ 1. Hence,
Pfn=Hn
n
∑
j=1
K
(n)
jn
ψn+j− 1
=Hn
n
∑
j=1
kjnK
(n)
jn
=HnKn,
which completes the proof of Theorem 5.1.
5.2.4 Three Further Theorems
The principal theorem, when expressed in the form
2 n− 1
∑
i=1
(−1)
i+1
Pf
(n)
i
ai, 2 n=HnKn, (5.2.23)
yields two corollaries by partial differentiation. Since the only elements
in Pfnwhich containφ 2 n− 1 andψ 2 n− 1 areai, 2 n,1≤i≤ 2 n−1, and
Pf
(n)
i
is independent ofai, 2 n, it follows that Pf
(n)
i
is independent ofφ 2 n− 1
andψ 2 n− 1. Moreover, these two functions occur only once inHnandKn,
respectively, both in position (n, n).