Page 200 - Determinants and Their Applications in Mathematical Physics

5.2 The Generalized Cusick Identities 185

From (5.2.4),

= ψ i .
∂a i,2n
∂φ 2n−1
Also,
= H n−1 .
∂H n
∂φ 2n−1
Hence,
2n−1
i+1 (n)
(−1) Pf ψ i = H n−1 K n . (5.2.24)
i
i=1
Similarly,
2n−1
i+1 (n)
(−1) Pf φ i = H n K n−1 . (5.2.25)
i
i=1
The following three theorems express modified forms of |a ij | n in terms of
the Hankelians.
Let B n (φ) denote the determinant which is obtained from |a ij | n by
replacing the last row by the row

φ 1 φ 2 φ 3 ...φ n .
Theorem 5.6.
2
a. B 2n−1 (φ)= H n−1 H n K n−1 ,
2
b. B 2n−1 (ψ)= H n−1 K n−1 K n ,
2
c. B 2n (φ)= −H K n−1 K n ,
n 2
d. B 2n (ψ)= −H n−1 H n K .
n
Proof. Expanding B 2n−1 (φ) by elements from the last row and their
cofactors and referring to (5.2.13), (5.2.14), and (5.2.25),
2n−1
(2n−1)
B 2n−1 (φ)= φ j A
2n−1,j
j=1
2n−1
(n) i+1 (n)
=Pf (−1) Pf
2n−1 i φ i
i=1
=Pf n−1 H n K n−1 . (5.2.26)
Part (a) now follows from Theorem 5.1 and (b) is proved in a similar
manner.
Expanding B 2n (φ) with the aid of Theorem 3.9 on bordered determinants
(Section 3.7) and referring to (5.2.11) and (5.2.25),
2n−1 2n−1
(2n−1)
B 2n (φ)= − a i,2n φ j A
ij
i=1 j=1