Page 163 - Determinants and Their Applications in Mathematical Physics

148 4. Particular Determinants

Theorem 4.52.
2

(xG ) = K n (h)x P (x, h),
h
n n
where
D h+n [x (1 + x) h+n−1 ]
n
P n (x, h)= .
(h + n − 1)!
Proof. Referring to (4.10.8),
K (n) h+i+j−1
x
n
n
G n (x)= ij
h + i + j − 1
i=1 j=1
n n h+i+j−1
V ni V nj x
= K n (h) . (4.11.57)
(h + i + j − 1) 2
i=1 j=1
Hence,
n n
i+j−2
(xG ) = K n (h)x h V ni V nj x

n
i=1 j=1
2
= K n (h)x P (x, h), (4.11.58)
h
n
where
n
i−1
P n (x, h)= (−1) n+i V ni x
i=1
n i−1
(h + n + i − 1)!x
=
(i − 1)! (n − i)! (h + i − 1)!
i=1
n h+n+i−1
D h+n (x )
= , (4.11.59)
(n − i)! (h + i − 1)!
i−1
n
h + n − 1 h+n+i−1
(h + n − 1)! P n (x, h)= D h+n (x )
h + i − 1
i=1

n
h + n − 1 h+i−1
= D h+n x n x
h + i − 1
i=1

h+n−1
h + n − 1
= D h+n x n x r
r
r=h
h+n−1
= D h+n x (1 + x) − p h+n−1 (x) , (4.11.60)
n
where p r (x) is a polynomial of degree r. The theorem follows.
Let
E(x)= |e ij (x)| n−1 ,