Determinants and Their Applications in Mathematical Physics

4.9 Hankelians 2 115

4.9 Hankelians 2

4.9.1 The Derivatives of Hankelians with Appell Elements

The Appell polynomial

φm=

m

∑

r=0

(

m

r

)

αrx

m−r

(4.9.1)

and other functions which satisfy the Appell equation

φ

′

m=mφm−^1 ,m=1,^2 ,^3 ,... , (4.9.2)

play an important part in the theory of Hankelians. Extensive notes on

these functions are given in Appendix A.4.

Theorem 4.33. If

An=|φm|n, 0 ≤m≤ 2 n− 2 ,

whereφmsatisfies the Appell equation, then

A

′

n

=φ

′

0

A

(n)

11

.

Proof. Split off them= 0 term from the double sum in relation (A 1 )in

Section 4.8.7:

A

′

A

=φ

′

0

∑

p+q=2

A

pq

+

2 n− 2

∑

m=1

φ

′

m

∑

p+q=m+2

A

pq

=φ

′

0 A

11

+

2 n− 2

∑

m=1

mφm− 1

∑

p+q=m+2

A

pq

.

The theorem follows from (E 1 ) and remains true if the Appell equation is

generalized to

φ

′

m

=mF φm− 1 ,F=F(x). (4.9.3)

Corollary. Ifφmis an Appell polynomial, thenφ 0 =α 0 =constant,A

′

=

0 , and, hence,Ais independent ofx, that is,

|φm(x)|n=|φm(0)|n=|αm|n, 0 ≤m≤ 2 n− 2. (4.9.4)

This identity is one of a family of identities which appear in Section 5.6.2

on distinct matrices with nondistinct determinants.

Ifφmsatisfies (4.9.3) andφ 0 = constant, it does not follows thatφmis

an Appell polynomial. For example, if

φm=(1−x

2

)

−m/ 2

Pm,