Determinants and Their Applications in Mathematical Physics

4.7 Wronskians 103

Then,

ε

′

rs=εr+1,s+εr,s+1 (4.7.15)

and

εr 0 =δr,n− 1. (4.7.16)

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

εrs=

{

0 ,r+s<n− 1

(−1)

s

,r+s=n−1.

(4.7.17)

Hence,

W(y 1 ,y 2 ,...,yn)W(W

1 n

,W

2 n

,...,W

nn

)

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

y 1 y 2 ··· yn

y

′

1 y

′

2 ··· y

′

n

y

(n−1)

1 y

(n−1)

2 ··· y

(n−1)

n

∣ ∣ ∣ ∣ ∣ ∣ ∣ n

∣ ∣ ∣ ∣ ∣ ∣ ∣

W

1 n

(W

1 n

)

′

··· (W

1 n

)

(n−1)

W

2 n

(W

2 n

)

′

··· (W

2 n

)

(n−1)

W

nn

(W

nn

)

′

··· (W

nn

)

(n−1)

∣ ∣ ∣ ∣ ∣ ∣ ∣ 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

εn− 2 , 0 εn− 2 , 1 ···

εn− 1 , 0 ···

∣

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

. (4.7.18)

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 ofn. The theorem follows.

The set of functions{W

1 n

,W

2 n

,...,W

nn

}are said to be adjunct to the

set{y 1 ,y 2 ,...,yn}.

Exercise.Prove that

W(y 1 ,y 2 ,...,yn)W(W

r+1,n

,W

r+2,n

,...,W

nn

)=W(y 1 ,y 2 ,...,yr),

1 ≤r≤n− 1 ,

by raising the order of the second Wronskian from (n−r)tonin a manner

similar to that employed in the section of the Jacobi identity.

4.7.6 Two-Way Wronskians.................

Let

Wn=|f

(i+j−2)

|n=|D

i+j− 2

f|n,D=

d

dx

,