Page 145 - Determinants and Their Applications in Mathematical Physics

130 4. Particular Determinants

Referring to the section on differences in Appendix A.8,

m
φ m =∆ θ 0
so that
B = A.
The Hankelian B arises in studies by M. Yamazaki and Hori of the Ernst
equation of general relativity and A arises in a related paper by Vein.
Define determinants U(x), V (x), and W, each of order (n + 1), by bor-
dering A in different ways. Since a ij is a function of x and y, it follows that
U(x) and V (x) are also functions of y. The argument x in U(x) and V (x)
refers to the variable which appears explicitly in the last row or column.
x

x /3
3

x /5
5
[a ij ] n
U(x)=

···

x
2n−1
/(2n − 1)
111 ··· 1 •

n+1
n n 2r−1
A rs x
= − , (4.10.24)
2r − 1
r=1 s=1
1

1/3

1/5

[a ij ] n
V (x)=
···

1/(2n − 1)
xx x ··· x •
3 5 2n−1
n+1
n n 2s−1
A rs x
= − , (4.10.25)
2r − 1
r=1 s=1
W = U(1) = V (1). (4.10.26)
Theorem 4.39.
2
2
2
2
2
p U (x)+ q U (y)= W − AW.
Proof.
A is x 2i−1 A jr x 2j−1
n
2

U (x)=
2i − 1 2j − 1
i,s j,r
A is A jr x 2(i+j−1)
= .
(2i − 1)(2j − 1)
i,j,r,s
Hence,
2
2
2
2
p U (x)+ q U (y) − W 2