Determinants and Their Applications in Mathematical Physics

74 4. Particular Determinants

The coefficient ofar, 2 n,1≤r≤(2n−1), in Pfnis found by putting

(is,js)=(r, 2 n) for any value ofs. Chooses= 1. Then, the coefficient is

∑

σrai 2 j 2 ai 3 j 3 ···ainjn,

where

σr= sgn

{

12 34...(2n−1) 2n

r 2 ni 2 j 2 ... in jn

}

2 n

= sgn

{

1234 ...(2n−1) 2n

ri 2 j 2 i 3 ... jn 2 n

}

2 n

= sgn

{

1234 ...(2n−1)

ri 2 j 2 i 3 ... jn

}

2 n− 1

(4.3.18)

=(−1)

r+1

sgn

{

1234 ...(r−1)r(r+1)...(2n−1)

i 2 j 2 i 3 j 3 ... r ... jn

}

2 n− 1

,r> 1

=(−1)

r+1

sgn

{

1234 ...(r−1)(r+1)...(2n−1)

i 2 j 2 i 3 j 3 ... ... ... jn

}

2 n− 2

,r> 1.

From (4.3.18),

σ 1 = sgn

{

1234 ... (2n−1)

1 i 2 j 2 i 3 ... jn

}

2 n− 1

= sgn

{

234 ... (2n−1)

i 2 j 2 i 3 ... jn

}

2 n− 2

.

Hence,

Pfn=

2 n− 1

∑

r=1

(−1)

r+1

ar, 2 nPf

(n)

r , (4.3.19)

where

Pf

(n)

r

=

∑

sgn

{

1234 ···(r−1)(r+1)···(2n−2) (2n−1)

i 2 j 2 i 3 j 3 ··· ··· ··· in jn

}

2 n− 2

×ai 2 j 2 ai 3 j 3 ···ainjn, 1 <r≤ 2 n− 1 , (4.3.20)

which is a Pfaffian of order (n−1) in which no element contains the row

parameterror the column parameter 2n. In particular,

Pf

(n)

2 n− 1

=

∑

sgn

{

1234 ··· (2n−3) (2n−2)

i 2 j 2 i 3 j 3 ··· in jn

}

2 n− 2

=Pfn− 1. (4.3.21)

Thus, a Pfaffian of orderncan be expressed as a linear combination of

(2n−1) Pfaffians of order (n−1).