Page 157 - Determinants and Their Applications in Mathematical Physics

142 4. Particular Determinants

Theorem 4.45 now follows from (4.11.24).
Theorem 4.46.
G n =(−1) n−1 v nn K n H n H n−1 ,
where G n is defined in (4.11.10).

Proof. Perform the row operation
n

R =

i R k
k=1
on H n and refer to the lemma. Row i becomes
3 5 2n−1
(x + t), (x + t), (x + t),..., (x + t) .
Hence,
n
2j−1 (n)
H n = (x + t)H , 1 ≤ i ≤ n. (4.11.26)
ij
j=1
It follows from the corollary to Theorem 4.44 that
n n
(n) (n) (n) (n) (n)
B = B = H H K . (4.11.27)
jr rs
ij ji si
r=1 s=1
Hence, applying (4.11.7),
(n) (n)
v ns H rs H
n
n
(n)
B jr . (4.11.28)
i + s − 1
ij = K n v ni
r=1 s=1
Put i = n, substitute the result into (4.11.10), and apply (4.11.16) and
(4.11.24):
(n)
n n n
v ns H rs 2j−1 (n)
(x + t)H
n + s − 1 jr
G n = K n v nn
r=1 s=1 j=1
(n)
n n
v ns H rs
n + s − 1
= K n v nn H n
r=1 s=1
= −K n v nn H n E n+1 . (4.11.29)
The theorem follows from Theorem 4.45.
4.11.3 Some Determinants with Binomial and Factorial
Elements
Theorem 4.47.

n + j − 2
n(n−1)/2
a. =(−1) ,
n − i

n