130 4. Particular Determinants
Referring to the section on differences in Appendix A.8,
φm=∆
m
θ 0
so that
B=A.
The HankelianBarises in studies by M. Yamazaki and Hori of the Ernst
equation of general relativity andAarises in a related paper by Vein.
Define determinantsU(x),V(x), andW, each of order (n+ 1), by bor-
deringAin different ways. Sinceaijis a function ofxandy, it follows that
U(x) andV(x) are also functions ofy. The argumentxinU(x) andV(x)
refers to the variable which appears explicitly in the last row or column.
U(x)=
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
∣
x
x
3
/ 3
[aij]n x
5
/ 5
···
x
2 n− 1
/(2n−1)
111 ··· 1 •
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
∣
n+1
=−
n
∑
r=1
n
∑
s=1
Arsx
2 r− 1
2 r− 1
, (4.10.24)
V(x)=
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
1
1 / 3
[aij]n 1 / 5
···
1 /(2n−1)
xx
3
x
5
··· x
2 n− 1
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
n+1
=−
n
∑
r=1
n
∑
s=1
Arsx
2 s− 1
2 r− 1
, (4.10.25)
W=U(1) =V(1). (4.10.26)
Theorem 4.39.
p
2
U
2
(x)+q
2
U
2
(y)=W
2
−AW.
Proof.
U
2
(x)=
∑
i,s
Aisx
2 i− 1
2 i− 1
n
∑
j,r
Ajrx
2 j− 1
2 j− 1
=
∑
i,j,r,s
AisAjrx
2(i+j−1)
(2i−1)(2j−1)
.
Hence,
p
2
U
2
(x)+q
2
U
2
(y)−W
2