Page 118 - Determinants and Their Applications in Mathematical Physics

4.7 Wronskians 103

Then,
ε = ε r+1,s + ε r,s+1 (4.7.15)
rs
and
ε r0 = δ r,n−1 . (4.7.16)

Differentiating (4.7.16) repeatedly and applying (4.7.15), it is found that

0, r + s<n − 1
ε rs = (4.7.17)
(−1) ,r + s = n − 1.
s
Hence,
W(y 1 ,y 2 ,...,y n )W(W 1n ,W 2n ,...,W nn )
)
)
W 1n (W 1n (W 1n (n−1)

y 1 y 2 ··· ···
y n
y y ··· y W (W ) ··· (W )
2n 2n 2n (n−1)
1 2 n

=
........................... ................................
(n−1) (n−1) (n−1)
)
y y W nn (W nn ··· (W nn (n−1)
)
1 2 ··· y n n n

ε 00 ε 01 ε 02 ··· ··· ε 0,n−2 ε 0,n−1

ε 10 ε 11 ε 12 ··· ε 1,n−3 ε 1,n−2

ε 20 ε 21 ε 22 ··· ε 2,n−3
. (4.7.18)
=
.................................................

ε n−2,0 ε n−2,1 ···

ε n−1,0 ···
n
From (4.7.17), it follows that those elements which lie above the secondary
diagonal are zero: those on the secondary diagonal from bottom left to top
right are
1, −1, 1,..., (−1) n+1
and the elements represented by the symbol are irrelevant to the value of
the determinant, which is 1 for all values of n. The theorem follows.
The set of functions {W 1n ,W 2n ,...,W nn } are said to be adjunct to the
set {y 1 ,y 2 ,...,y n }.
Exercise. Prove that
W(y 1 ,y 2 ,...,y n )W(W r+1,n ,W r+2,n ,...,W nn )= W(y 1 ,y 2 ,...,y r ),
1 ≤ r ≤ n − 1,
by raising the order of the second Wronskian from (n−r)to n in a manner
similar to that employed in the section of the Jacobi identity.
4.7.6 Two-Way Wronskians
Let
d
(i+j−2)
i+j−2
W n = |f | n = |D f| n , D = ,
dx