Determinants and Their Applications in Mathematical Physics

168 4. Particular Determinants

Define three other matricesM

′

,K

′

, andN

′

of ordernas follows:

M

′

=[α

′

ij]n (symmetric),

K

′

=[2

i+j− 1

(ki+j+ki+j− 2 )]n (Hankel),

N

′

=[β

′

ij

]n (lower triangular),

(4.13.14)

wherekris defined in (4.13.5);

α

′

ij

=

{

(−1)

j− 1

ui−j+ui+j,j≤i

(−1)

i− 1

uj−i+ui+j,j≥i,

(4.13.15)

β

′

ij=0,j>iori+jodd,

β

′

2 i, 2 j=

1

2

μij, 1 ≤j≤i,

β

′

2 i+1, 2 j+1=λij+

1

2

μij, 0 ≤j≤i. (4.13.16)

The functionsλijand

1

2

μijappear in Appendix A.10.μij=(2j/i)λij.

Theorem 4.61.

M=N

′

K(N

′

)

T

.

The details of the proof are similar to those of Theorem 4.60.

Let

N

′

K

′

(N

′

)

T

=[γ

′

ij]n

and consider the four cases separately. It is found with the aid of

Theorem A.8(e) in Appendix A.10 that

γ

′

2 p+1, 2 q+1

=

N

∑

j=1

aij

{

gq−p(xj)+gq+p+1(xj)

}

=α

′

2 p+1, 2 q+1

(4.13.17)

and further thatγ

′

ij=α

′

ijfor all values ofiandj.

Corollary.

|α

′

ij|n=|M

′

|n=|N

′

|

2

n|K

′

|n

=|β

′

ij|

2

n^2

n

∣

∣ 2 i+j−^2 (k

i+j+ki+j− 2 )

∣

∣

n

=2

n

2

|ki+j+ki+j− 2 |n (4.13.18)

sinceβ

′

ii= 1 for all values ofi. Thus,M

′

can also be expressed as a

Hankelian.

4.13.2 The Factors of a Particular Symmetric Toeplitz

Determinant

The determinants

Pn=

1

2

|pij|n,