Page 122 - Determinants and Their Applications in Mathematical Physics

4.8 Hankelians 1 107

The second proof illustrates the equivalence of row and column op-
erations on the one hand and matrix-type products on the other
(Section 2.3.2).
Second Proof. Define a triangular matrix P(x) as follows:

i − 1
P(x)= x i−j
j − 1
n
1
 
 x 1 
=  x 2 2x 1  . (4.8.13)


x 3x 3x 1
 3 2 
................
n
Since |P(x)| = |P (x)| = 1 for all values of x.
T
T
A = |P(−h)AP (−h)| n

i − 1 j − 1
= (−h) i−j (−h) j−i

j − 1 |φ i+j−2 | n i − 1
n n
(4.8.14)
= |α ij | n
where, applying the formula for the product of three determinants at the
end of Section 3.3.5,

i j
i − 1 j − 1
α ij = (−h) i−r φ r+s−2 (−h) j−s
r − 1 s − 1
r=1 s=1
i−1
j−1
i − 1 j − 1

= (−h) i−1−r (−h) j−1−s
r s φ r+s
r=0 s=0
i−1
i − 1 j−1

= (−h) i−1−r ∆
r h φ r
r=0
i−1
j−1 i − 1
=∆ (−h) i−1−r
h r φ r
r=0
=∆ j−1 ∆ i−1 φ 0
h h
=∆ i+j−2 φ 0 . (4.8.15)
n
The theorem follows. Simple differences are obtained by putting
h =1.
Exercise. Prove that
n n
r+s−2
h A rs (x)= A 11 (x − h).
r=1 s=1