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 matrixP(x) as follows:
P(x)=
[(
i− 1
j− 1
)
x
i−j
]
n
=
1
x 1
x
2
2 x 1
x
3
3 x
2
3 x 1
................
n
. (4.8.13)
Since|P(x)|=|P
T
(x)|= 1 for all values ofx.
A=|P(−h)AP
T
(−h)|n
=
∣
∣
∣
∣
(−h)
i−j
(
i− 1
j− 1
)∣
∣
∣
∣
n
|φi+j− 2 |n
∣
∣
∣
∣
(−h)
j−i
(
j− 1
i− 1
)∣
∣
∣
∣
n
=|αij|n (4.8.14)
where, applying the formula for the product of three determinants at the
end of Section 3.3.5,
αij=
i
∑
r=1
j
∑
s=1
(−h)
i−r
(
i− 1
r− 1
)
φr+s− 2 (−h)
j−s
(
j− 1
s− 1
)
=
i− 1
∑
r=0
(
i− 1
r
)
(−h)
i− 1 −r
j− 1
∑
s=0
(
j− 1
s
)
(−h)
j− 1 −s
φr+s
=
i− 1
∑
r=0
(
i− 1
r
)
(−h)
i− 1 −r
∆
j− 1
h
φr
=∆
j− 1
h
i− 1
∑
r=0
(
i− 1
r
)
(−h)
i− 1 −r
φr
=∆
j− 1
h
∆
i− 1
h
φ 0
=∆
i+j− 2
n φ^0. (4.8.15)
The theorem follows. Simple differences are obtained by putting
h=1.
Exercise.Prove that
n
∑
r=1
n
∑
s=1
h
r+s− 2
Ars(x)=A 11 (x−h).