Determinants and Their Applications in Mathematical Physics

5.1 Determinants Which Represent Particular Polynomials 171

b.ψn(x)=

1

n!

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

α 0 α 1 α 2 α 3 ··· αn− 1 αn

nx

n− 12 x

n− 23 x

.....................

1 nx

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

.

Both determinants are Hessenbergians (Section 4.6).

Proof of (a).Denote the determinant byHn+1, expand it by the two

elements in the last row, and repeat this operation on the determinants of

lower order which appear. The result is

Hn+1(x)=

n

∑

r=1

(

n

r

)

Hn+1−r(−x)

r

+(−1)

n

αn.

TheHn+1term can be absorbed into the sum, giving

(−1)

n

αn=

n

∑

r=0

(

n

r

)

Hn+1−r(−x)

r

.

This is an Appell polynomial whose inverse relation is

Hn+1(x)=

n

∑

r=0

(

n

r

)

(−1)

n−r

αn−rx

r

,

which is equivalent to the stated result.

Proof of (b).Denote the determinant byH

∗

n+1

and note that some

of its elements are functions ofn, so that the minor obtained by removing

its last row and column isnotequal toH

∗

nand hence there is no obvious

recurrence relation linkingH

∗

n+1,H

∗

n,H

∗

n− 1 , etc.

The determinantH

∗

n+1can be obtained by transformingHn+1by a series

of row operations which reduce some of its elements to zero. MultiplyRi

by (n+2−i), 2≤i≤n+ 1, and compensate for the unwanted factorn!by

dividing the determinant by that factor. Now perform the row operations

R

′

i=Ri−

(

i− 1

n+1−i

)

xRi+1

first with 2≤i≤n, which introduces (n−1) zero elements intoCn+1,

then with 2≤i≤n−1, which introduces (n−2) zero elements intoCn,

then with 2≤i≤n−2, etc., and, finally, withi= 2. The determinant

H
∗
n+1
appears.