Page 162 - Determinants and Their Applications in Mathematical Physics

4.11 Hankelians 4 147

Proof. By interchanging first rows and then columns in a suitable
manner it is easy to show that

ν 0 ν 1 ν 2 ···

ν 1 ν 2 ν 3 ···

ν 2 ν 3 ν 4 ···
E n = ............... . (4.11.53)

ν 1 ν 2
···

ν 2 ν 3
···
.............

n
Hence, referring to Theorems 4.11.5 and 4.11.6b,
(n+1)
E 2n =(−1) A n A
n
n+1,1
2
n −(2n−1)
=(−1) 2 ,
(n+1)
E 2n+1 =(−1) A n+1 A n+1,1
n
2
=(−1) 2 . (4.11.54)
n −4n
These two results can be combined into one as shown in the theorem which
is applied in Section 4.12.1 to evaluate |P m (x)| n .
Exercise. If

2m
B n = , 0 ≤ m ≤ 2n − 2,
m
n
prove that
B n =2 n−1 ,
(n) 2[n(n−1)−(i+j−2)] (n)
B =2 A ,
ij ij
(n) n−1
B n1 =2 .
4.11.4 A Nonlinear Differential Equation
Let
G n (x, h, k)= |g ij | n ,
where
h+i+k−1

,j = k
x
g ij = h+i+k−1 (4.11.55)
1
h+i+j−1 , j = k.
Every column in G n except column k is identical with the corresponding
column in the generalized Hilbert determinant K n (h). Also, let
n

G n (x, h)= G n (x, h, k). (4.11.56)
k=1