Page 117 - Determinants and Their Applications in Mathematical Physics

102 4. Particular Determinants
1 (n) (n)
= − W w r,s+1 W ,
W 2 is rj
n
s=j−1,n r

(n) (n) (n) (n) (n+1)

W n W − W W = −W n W + W W .
n i,j−1
ij ij in n+1,j
Hence, referring to (4.7.7) and Theorem 4.26(a),
(n) (n+1) (n) (n+1) (n) (n)
W W − W W = −W n (W ) + W i,j−1
ij n+1,n in n+1,j ij
(n+1)
= W n W ,
i,n+1;jn
which proves Theorem 4.27.
4.7.4 An Arbitrary Determinant
Since the functions y i are arbitrary, we may let y i be a polynomial of degree
(n − 1). Let
a ir x
n r−1

y i = , (4.7.12)
(r − 1)!
r=1
where the coefficients a ir are arbitrary. Furthermore, since x is arbitrary,
we may let x = 0 in algebraic identities. Then,
(j−1)
w ij = y (0)
i
= a ij . (4.7.13)
Hence, an arbitrary determinant A n = |a ij | n can be expressed in the
form (W n ) x=0 and any algebraic identity which is satisfied by an arbitrary
Wronskian is valid for A n .

4.7.5 Adjunct Functions

Theorem.
W(y 1 ,y 2 ,...,y n )W(W 1n ,W 2n ,...,W nn )=1.
Proof. Since

(n−2) 0, 0 ≤ r ≤ n − 2

CC C ··· C C (r) =
W, r = n − 1,
it follows by expanding the determinant by elements from its last column
and scaling the cofactors that
n
(r)
y W in = δ r,n−1 .
i
i=1
Let
n
(r) in (s)
ε rs = y (W ) . (4.7.14)
i
i=1