102 4. Particular Determinants
=−
1
W
2
n
∑
s=j− 1 ,n
W
(n)
is
∑
r
wr,s+1W
(n)
rj
,
Wn
(
W
(n)
ij
)′
−W
(n)
ij
W
′
n=−WnW
(n)
i,j− 1
+W
(n)
in
W
(n+1)
n+1,j
Hence, referring to (4.7.7) and Theorem 4.26(a),
W
(n)
ij W
(n+1)
n+1,n−W
(n)
inW
(n+1)
n+1,j=−Wn
[
(W
(n)
ij )
′
+W
(n)
i,j− 1
]
=WnW
(n+1)
i,n+1;jn
,
which proves Theorem 4.27.
4.7.4 An Arbitrary Determinant
Since the functionsyiare arbitrary, we may letyibe a polynomial of degree
(n−1). Let
yi=
n
∑
r=1
airx
r− 1
(r−1)!
, (4.7.12)
where the coefficientsairare arbitrary. Furthermore, sincexis arbitrary,
we may letx= 0 in algebraic identities. Then,
wij=y
(j−1)
i (0)
=aij. (4.7.13)
Hence, an arbitrary determinant An = |aij|ncan be expressed in the
form (Wn)x=0and any algebraic identity which is satisfied by an arbitrary
Wronskian is valid forAn.
4.7.5 Adjunct Functions...................
Theorem.
W(y 1 ,y 2 ,...,yn)W(W
1 n
,W
2 n
,...,W
nn
)=1.
Proof. Since
∣
∣
CC
′
C
′′
···C
(n−2)
C
(r)
∣
∣
=
{
0 , 0 ≤r≤n− 2
W, r=n−1,
it follows by expanding the determinant by elements from its last column
and scaling the cofactors that
n
∑
i=1
y
(r)
i
W
in
=δr,n− 1.
Let
εrs=
n
∑
i=1
y
(r)
i
(W
in
)
(s)
. (4.7.14)