Determinants and Their Applications in Mathematical Physics

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).