Page 225 - Determinants and Their Applications in Mathematical Physics

210 5. Further Determinant Theory

c 1 c 2 c 3 ··· c n+1

c 2 c 3 c 4 ··· c n+2
b. Q 2n = 1 ................................... .
B n
c n+1 c n+2 ··· c 2n
c n
ψ 0 x n ψ 1 x ψ 2 x ···
n−1 n−2
n+1
ψ n
Proof. From the second equation in (5.5.24) in the previous section and
referring to Theorem 5.11a,
r

q 2n−1,r = c r−t p 2n−1,t , 0 ≤ r ≤ n − 1
t=0
1 (n+1)
r
= c r−t A .
n+1,n+1−t
t=0
A n
Hence, from the second equation in (5.5.21) with n → n − 1 and applying
(n+1)
Lemma (a) with u r → x and v s → A ,
r
n+1,s
n−1 r
(n+1)
A n Q 2n−1 = x r c r−t A
n+1,n+1−t
r=0 t=0
j−1
n
(n+1)
= A c r x n+r−j
n+1,j+1
j=1 r=0
j−1
n
(n+1)
= x n−j A n+1,j+1 c r x r
j=1 r=0
n
(n+1)

= ψ j−1 x n−j A n+1,j+1 .
j=1
This sum represents a determinant of order (n + 1) whose first n rows are
identical with the first n rows of the determinant in part (a) of the theorem
and whose last row is
n−1 n−2 n−3
0 ψ 0 x ψ 1 x ψ 2 x ··· ψ n−1 .
n+1
The proof of part (a) is completed by performing the row operation
R n+1 = R n+1 + x R 1 .
n
The proof of part (b) of the theorem applies Lemma (b) and gives the
required result directly, that is, without the necessity of performing a
row operation. From (5.5.27) in the previous section and referring to
Theorem 5.11b,
r

q 2n,r = c r−t p 2n,t , 0 ≤ r ≤ n
t=0