Page 183 - Determinants and Their Applications in Mathematical Physics

168 4. Particular Determinants

Define three other matrices M , K , and N of order n as follows:

(symmetric),
M =[α ] n
ij

K =[2 i+j−1 (k i+j + k i+j−2 )] n (Hankel), (4.13.14)

(lower triangular),
N =[β ] n
ij
where k r is defined in (4.13.5);
j−1
(−1) u i−j + u i+j ,j ≤ i

α = i−1 (4.13.15)
(−1) u j−i + u i+j , j ≥ i,
ij

β =0, j > i or i + j odd,
ij
1
β = µ ij , 1 ≤ j ≤ i,
2
2i,2j
1
β = λ ij + µ ij , 0 ≤ j ≤ i. (4.13.16)
2i+1,2j+1 2
1
The functions λ ij and µ ij appear in Appendix A.10. µ ij =(2j/i)λ ij .
2
Theorem 4.61.

M = N K(N ) .
T
The details of the proof are similar to those of Theorem 4.60.
Let
T
N K (N ) =[γ ] n
ij
and consider the four cases separately. It is found with the aid of
Theorem A.8(e) in Appendix A.10 that
N
- .
γ 2p+1,2q+1 = a ij g q−p (x j )+ g q+p+1 (x j )
j=1

= α 2p+1,2q+1 (4.13.17)

and further that γ = α for all values of i and j.
ij ij
Corollary.
2
|α | n = |M | n = |N | |K | n
ij n
2 n i+j−2

= |β | 2 2 (k i+j + k i+j−2 )
ij n
n
2
(4.13.18)
n
=2 |k i+j + k i+j−2 | n

since β = 1 for all values of i. Thus, M can also be expressed as a
ii
Hankelian.
4.13.2 The Factors of a Particular Symmetric Toeplitz
Determinant
The determinants
1
P n = |p ij | n ,
2