Determinants and Their Applications in Mathematical Physics

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