Page 181 - Determinants and Their Applications in Mathematical Physics

166 4. Particular Determinants

β 2i,2i = λ ii , i ≥ 1,
β 2i+1,2j+1 = λ ij , 0 ≤ j ≤ i,
β 2i+2,2j = λ i+1,j − λ ij , 1 ≤ j ≤ i +1, (4.13.6)
i i + j
λ ij = ;
i + j 2j
1
λ ii = , i > 0; λ i0 =1,i ≥ 0. (4.13.7)
2
The functions λ ij and f r (x) appear in Appendix A.10.
Theorem 4.60.

M = NKN .
T
Proof. Let
G =[γ ij ] n = NKN . (4.13.8)
T
Then
G = NK N T
T
T
= NKN T
= G.
Hence, G is symmetric, and since M is also symmetric, it is sufficient to
prove that α ij = γ ij for j ≤ i. There are four cases to consider:

i. i, j both odd,
ii. i odd, j even,
iii. i even, j odd,
iv. i, j both even.
To prove case (i), put i =2p+1 and j =2q+1 and refer to Appendix A.10,
where the definition of g r (x) is given in (A.10.7), the relationships between
f r (x) and g r (x) are given in Lemmas (a) and (b) and identities among the
g r (x) are given in Theorem 4.61.

α 2p+1,2q+1 = u 2q+2p + u 2q−2p
N

- .
= a j f 2q+2p (x j )+ f 2q−2p (x j )
j=1
N

- .
= a j g q+p (x j )+ g q−p (x j )
j=1
N

=2 a j g p (x j )g q (x j ). (4.13.9)
j=1
It follows from (4.13.8) and the formula for the product of three matrices
(the exercise at the end of Section 3.3.5) with appropriate adjustments to