Page 198 - Determinants and Their Applications in Mathematical Physics

5.2 The Generalized Cusick Identities 183
n n+j−1
(n) (n) 2n−i−1
= H K x
jn i−j+1,n
j=1 i=j
2n−1 n
2n−i−1 (n) (n)
= x H K . (5.2.20)
jn i−j+1,n
i=1 j=1
Note that the changes in the limits of the i-sum have introduced only zero
terms. The lemma follows by equating coefficients of x 2n−i−1 .

5.2.3 Proof of the Principal Theorem
A double-sum identity containing the symbols c ij , f i , and g i is given in
Appendix A.3. It follows from Lemma 5.5 that the conditions defining the
validity of the double-sum identity are satisfied if

f i =(−1) i+1 Pf (n) ,
i
(n) (n)
c ij = H K ,
in jn
g i = a i,2n .
Hence,
2n−1 n n
i+1 (n) (n) (n)
(−1) Pf a i,2n = H K
i in jn a i+j−1,2n
i=1 i=1 j=1
2n−i−j+1
n n
(n) (n)
= H K φ s+i+j−2 ψ 2n−s .
in jn
i=1 j=1 s=1
From (5.2.11), the sum on the left is equal to Pf n . Also, since the interval
(1, 2n−i−j +1) can be split into the intervals (1,n+1−j) and (n+2−j,
2n − i − j + 1), it follows from the note in Appendix A.3 on a triple sum
that
n−1
n
(n) (n)
Pf n = K X j + H Y i ,
jn in
j=1 i=1
where
n n+1−j
(n)
X j = H
in φ s+i+j−2 ψ 2n−s
i=1 s=1
n+1−j
n
(n)
= φ s+i+j−2 H
ψ 2n−s
in
s=1 i=1
n+1−j n
(n)
= h i,s+j−1 H
ψ 2n−s
in
s=1 i=1