Page 89 - Determinants and Their Applications in Mathematical Physics

74 4. Particular Determinants

The coefficient of a r,2n ,1 ≤ r ≤ (2n − 1), in Pf n is found by putting
(i s ,j s )=(r, 2n) for any value of s. Choose s = 1. Then, the coefficient is

a ,
σ r a i 2 j 2 i 3 j 3 ··· a i n j n
where

12 3 4 ... (2n − 1) 2n
σ r = sgn
r 2ni 2 j 2 ... i n j n
2n

1 234 ... (2n − 1) 2n
= sgn
ri 2 j 2 i 3 ... j n 2n
2n

1 2 3 4 ... (2n − 1)
= sgn (4.3.18)
ri 2 j 2 i 3 ...
2n−1
j n

1234 ... (r − 1)r(r +1) ... (2n − 1)
r+1
=(−1) sgn ,r > 1
i 2 j 2 i 3 j 3 ... r ...
2n−1
j n

1234 ... (r − 1)(r +1) ... (2n − 1)
r+1
=(−1) sgn ,r > 1.
i 2 j 2 i 3 j 3 ... ... ...
2n−2
j n
From (4.3.18),

1 2 3 4 ... (2n − 1)
σ 1 = sgn
1 i 2 j 2 i 3 ...
2n−1
j n

2 3 4 ... (2n − 1)
= sgn .
i 2 j 2 i 3 ...
2n−2
j n
Hence,
2n−1
r+1 (n)
Pf n = (−1) a r,2n Pf , (4.3.19)
r
r=1
where

1234 ··· (r − 1)(r +1) ··· (2n − 2) (2n − 1)
(n)

Pf = sgn
r
i 2 j 2 i 3 j 3 ··· ··· ···
i n
j n
2n−2
a , 1 <r ≤ 2n − 1, (4.3.20)
×a i 2 j 2 i 3 j 3 ··· a i n j n
which is a Pfaffian of order (n − 1) in which no element contains the row
parameter r or the column parameter 2n. In particular,

(n) 1 2 3 4 ··· (2n − 3) (2n − 2)
Pf 2n−1 = sgn
i 2 j 2 i 3 j 3 ···
2n−2
j n
i n
=Pf n−1 . (4.3.21)
Thus, a Pfaffian of order n can be expressed as a linear combination of
(2n − 1) Pfaffians of order (n − 1).