5.2 The Generalized Cusick Identities 183
=
n
∑
j=1
n+j− 1
∑
i=j
H
(n)
jnK
(n)
i−j+1,nx
2 n−i− 1
=
2 n− 1
∑
i=1
x
2 n−i− 1
n
∑
j=1
H
(n)
jn
K
(n)
i−j+1,n
. (5.2.20)
Note that the changes in the limits of thei-sum have introduced only zero
terms. The lemma follows by equating coefficients ofx
2 n−i− 1
.
5.2.3 Proof of the Principal Theorem
A double-sum identity containing the symbolscij,fi, andgiis 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
fi=(−1)
i+1
Pf
(n)
i
,
cij=H
(n)
in
K
(n)
jn
,
gi=ai, 2 n.
Hence,
2 n− 1
∑
i=1
(−1)
i+1
Pf
(n)
i
ai, 2 n=
n
∑
i=1
n
∑
j=1
H
(n)
in
K
(n)
jn
ai+j− 1 , 2 n
=
n
∑
i=1
n
∑
j=1
H
(n)
in
K
(n)
jn
2 n−i−j+1
∑
s=1
φs+i+j− 2 ψ 2 n−s.
From (5.2.11), the sum on the left is equal to Pfn. Also, since the interval
(1, 2 n−i−j+ 1) can be split into the intervals (1,n+1−j) and (n+2−j,
2 n−i−j+ 1), it follows from the note in Appendix A.3 on a triple sum
that
Pfn=
n
∑
j=1
K
(n)
jn
Xj+
n− 1
∑
i=1
H
(n)
in
Yi,
where
Xj=
n
∑
i=1
H
(n)
in
n+1−j
∑
s=1
φs+i+j− 2 ψ 2 n−s
=
n+1−j
∑
s=1
ψ 2 n−s
n
∑
i=1
φs+i+j− 2 H
(n)
in
=
n+1−j
∑
s=1
ψ 2 n−s
n
∑
i=1
hi,s+j− 1 H
(n)
in