Determinants and Their Applications in Mathematical Physics

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