Determinants and Their Applications in Mathematical Physics

298 6. Applications of Determinants in Mathematical Physics

where

Em=

m

∏

r=1

εk

r

,

K=

m

∑

r=1

(kr−1). (6.10.39)

Applying Theorem (c) in Section 4.1.8 on Vandermondian identities,

Ym=(−1)

K

Em

V(ck

m+1

,ck

m+2

,...,ck

2 n

){V 2 n(c)}

m− 1

V(ck

1

,ck

2

,...,ck

m

)

. (6.10.40)

Hence,

Wm=

(−1)

K

{V 2 n(c)}

m− 1

ρ

m(m−1)

∑

1 ≤k 1 <k 2 <...<km≤ 2 n

·EmV(ck 1 ,ck 2 ,...,ckm)V(ckm+1,ckm+2,...,ck 2 n).(6.10.41)

Using the Laplace formula (Section 3.3) to expandH

(m)

2 n

(ε) by the first

mrows and the remaining (2n−m) rows and referring to the exercise at

the end of Section 4.1.8,

H

(m)

2 n

(ε)=

∑

1 ≤k 1 <k 2 <···<km≤ 2 n

N 12 ···m;k

1 ,k 2 ,...,km

A 12 ···m;k

1 ,k 2 ,...,km

,

(6.10.42)

where

N 12 ···m;k

1 ,k 2 ,...,km

=EmV(ck

1

,ck

2

,...,ck

m

),

A 12 ···m;k

1 ,k 2 ,...,km

=(−1)

R

M 12 ···m;k

1 ,k 2 ,...,km

=(−1)

R

V(ck

m+1

,ck

m+2

,...,ck

2 n

), (6.10.43)

whereMis the unsigned minor associated with the cofactorAandRis

the sum of their parameters. Referring to (6.10.39),

R=

1

2

m(m+1)+

m

∑

r=1

kr

=K+

1

2

m(m−1). (6.10.44)

Hence,

H

(m)

2 n

(ε)=(−1)

R

∑

1 ≤k 1 <k 2 <···<km≤ 2 n

EmV(ck

1

,ck

2

,...,ck

m

)

·V(ckm+1,ckm+2,...,ck 2 n)

=

(−ρ

2

)

m(m−1)/ 2

{V 2 n(c)}

m− 1

Wm, (6.10.45)

which proves part (a) of the theorem. Part (b) can be proved in a similar

manner.