Determinants and Their Applications in Mathematical Physics

4.7 Wronskians 101

c.

(

W

(n)

in

)

′

=−W

(n)

i,n− 1.

Proof. LetZidenote then-rowed column vector in which the element

in rowiis 1 and all the other elements are zero.

Then

W

(n)

ij

=

∣

∣

C 1 ···Cj− 2 Cj− 1 ZiCj+1···Cn− 1 Cn

∣

∣

n

, (4.7.10)

(

W

(n)

ij

)′

=

∣

∣C

1 ···Cj− 2 CjZiCj+1···Cn− 1 Cn

∣

∣

n

+

∣

∣C

1 ···Cj− 2 Cj− 1 ZiCj+1···Cn− 1 Cn+1

∣

∣

n

.(4.7.11)

Formula (a) follows afterCjandZiin 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.Wi 0 =0;Wrs,tt=0.

Lemma.When 1 ≤j, s≤n,

n

∑

r=0

wr,s+1W

(n)

rj =







Wn,s=j−1,j=1,

−W

(n+1)

n+1,j

,s=n,

0 , otherwise.

The first and third relations are statements of the sum formula for

elements and cofactors (Section 2.3.4):

n

∑

r=1

wr,n+1W

(n)

rj

=

∣

∣C

1 C 2 ···Cj− 1 Cn+1Cj+1···Cn

∣

∣

n

=(−1)

n−j

∣

∣C

1 C 2 ···Cj− 1 Cj+1···CnCn+1

∣

∣

n

.

The second relation follows.

Theorem 4.27.

∣

∣

∣

∣

∣

W

(n)

ij

W

(n)

in

W

(n+1)

n+1,j

W

(n+1)

n+1,n

∣

∣

∣

∣

∣

=WnW

(n+1)

i,n+1;jn

.

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),

(

W

ij

n

)′

=−

n

∑

r=1

n

∑

s=1

w

′

rsW

is

nW

rj

n.

Reverting to simple cofactors and applying the above lemma,

(

W

(n)

ij

Wn

)′

=−

1

W

2

n

∑

r

∑

s

w

′

rsW

(n)

is W

(n)

rj