Determinants and Their Applications in Mathematical Physics

4.11 Hankelians 4 145

where

C

′

n=λn−^1 ,n−^1 Cn+

n− 1

∑

j=1

λn− 1 ,j− 1 Cj

=

n− 1

∑

j=1

λn− 1 ,j− 1

[

νj− 1 νj···νn+j− 3 νn+j− 2

]T

n

=2

−(4n−5)

[

00 ··· 01

]T

n

. (4.11.42)

Hence,

An=2

−4(n−1)+1

An− 1

An− 1 =2

−4(n−2)+1

An− 2

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

A 2 =2

−4(1)+1

A 1 , (A 1 =ν 0 =1).

(4.11.43)

The theorem follows by equating the product of the left-hand sides to the

product of the right-hand sides.

It is now required to evaluate the cofactors ofAn.

Theorem 4.49.

a.A

(n)

nj

=2

−(n−1)(2n−3)

λn− 1 ,j− 1 ,

b.A

(n)

n 1 =2

−(n−1)(2n−3)

,

c. A

nj

n =2

2(n−1)

λn− 1 ,j− 1.

Proof. Thenequations in (4.11.40) can be expressed in matrix form as

follows:

AnLn=C

′

n, (4.11.44)

where

Ln=

[

λn 0 λn 1 ···λn,n− 2 λn,n− 1

]

T

n

. (4.11.45)

Hence,

Ln=A

− 1

nC

′

n

=A

− 1

n

[

A

(n)

ji

]

n

C

′

n

=2

(n−1)(2n−1)−2(n−1)

[

An 1 An 2 ···An,n− 1 Ann

]T

n

, (4.11.46)

which yields part (a) of the theorem. Parts (b) and (c) then follow

easily.

Theorem 4.50.

A

(n)

ij =2

−n(2n−3)

[

2

2 i− 3

λi− 1 ,j− 1 +

n− 1

∑

r=i+1

λr− 1 ,i− 1 λr− 1 ,j− 1

]

,j≤i<n− 1.