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 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).