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,