Determinants and Their Applications in Mathematical Physics

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)