Determinants and Their Applications in Mathematical Physics

146 4. Particular Determinants

Proof. Apply the Jacobi identity (Section 3.6.1) toAr, wherer≥i+1:

∣

∣

∣

∣

∣

A

(r)

ij

A

(r)

ir

A

(r)

rj A

(r)

rr

∣

∣

∣

∣

∣

=ArA

(r)

ir,jr

,

=ArA

(r−1)

ij

,

Ar− 1 A

(r)

ij

−ArA

(r−1)

ij

=A

(r)

ir

A

(r)

jr

. (4.11.47)

Scale the cofactors and refer to Theorems 4.48 and 4.49a:

A

ij

r−A

ij

r− 1

=

Ar

Ar− 1

(

A

ri

r A

rj

r

)

=2

−(4r−5)

A

ri

rA

rj

r

=2λr− 1 ,i− 1 λr− 1 ,j− 1. (4.11.48)

Hence,

2

n

∑

r=i+1

λr− 1 ,i− 1 λr− 1 ,j− 1 =

n

∑

r=i+1

(

A

ij

r

−A

ij

r− 1

)

=A

ij

n−A

ij

i

=A

ij

n

− 2

2(i−1)

λi− 1 ,j− 1 , (4.11.49)

which yields a formula for the scaled cofactorA

ij

n. The stated formula for

the simple cofactorA

(n)

ij follows from Theorem 4.49a. 

Let

En=|Pm(0)|n, 0 ≤m≤ 2 n− 2 , (4.11.50)

wherePm(x) is the Legendre polynomial [Appendix A.5]. Then,

P 2 m+1(0)=0,

P 2 m(0) = νm. (4.11.51)

Hence,

En=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

ν 0 • ν 1 • ν 2 ···

• ν 1 • ν 2 • ···

ν 1 • ν 2 • ν 3 ···

• ν 2 • ν 3 • ···

ν 2 • ν 3 • ν 4 ···

........................

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

. (4.11.52)

Theorem 4.51.

En=|Pm(0)|n=(−1)

n(n−1)/ 2

2

−(n−1)

2

.