Page 116 - Determinants and Their Applications in Mathematical Physics

4.7 Wronskians 101

(n) (n)

c. W = −W i,n−1 .
in
Proof. Let Z i denote the n-rowed column vector in which the element
in row i is 1 and all the other elements are zero.
Then
(n)
W = C 1 ··· C j−2 C j−1 Z i C j+1 ··· C n−1 C n , (4.7.10)

ij
n

(n)
W
ij = C 1 ··· C j−2 C j Z i C j+1 ··· C n−1 C n
n

+ C 1 ··· C j−2 C j−1 Z i C j+1 ··· C n−1 C n+1 . (4.7.11)

n
Formula (a) follows after C j and Z i in the first determinant are inter-
changed. Formulas (b) and (c) are special cases of (a) which can be proved
by a similar method but may also be obtained from (a) by referring to the
definition of first and second cofactors. W i0 =0; W rs,tt =0.
Lemma. When 1 ≤ j, s ≤ n,

 W n , s = j − 1, j =1,
n
(n) (n+1)
w r,s+1 W = −W ,s = n,
rj n+1,j

r=0 0, otherwise.
The first and third relations are statements of the sum formula for
elements and cofactors (Section 2.3.4):
n
(n)
w r,n+1 W
rj = C 1 C 2 ··· C j−1 C n+1 C j+1 ··· C n
n
r=1

=(−1) n−j C 1 C 2 ··· C j−1 C j+1 ··· C n C n+1 .

n
The second relation follows.
Theorem 4.27.
(n)

W W (n) (n+1)

ij in = W n W .
(n+1) (n+1) i,n+1;jn
W W

n+1,j n+1,n
This identity is a particular case of Jacobi variant (B) (Section 3.6.3)
with (p, q) → (j, n), but the proof which follows is independent of the
variant.
Proof. Applying double-sum relation (B) (Section 3.4),
n n
ij

W = − w W W .

rj
is
n rs n n
r=1 s=1
Reverting to simple cofactors and applying the above lemma,
(n)
W 1 (n) (n)
ij
= − 2 w W W
W rs is rj
W n
n
r s