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4.9 Hankelians 2 115
4.9 Hankelians 2

4.9.1 The Derivatives of Hankelians with Appell Elements
The Appell polynomial

m
m

φ m = α r x m−r (4.9.1)
r
r=0
and other functions which satisfy the Appell equation
φ = mφ m−1 , m =1, 2, 3,... , (4.9.2)
m
play an important part in the theory of Hankelians. Extensive notes on
these functions are given in Appendix A.4.

Theorem 4.33. If
A n = |φ m | n , 0 ≤ m ≤ 2n − 2,
where φ m satisfies the Appell equation, then
(n)

A = φ A 11 .
0
n
Proof. Split off the m = 0 term from the double sum in relation (A 1 )in
Section 4.8.7:
2n−2
A
= φ 0 A pq + φ A pq
A m
p+q=2 m=1 p+q=m+2
2n−2
= φ A 11 + A .

pq
0 mφ m−1
m=1 p+q=m+2
The theorem follows from (E 1 ) and remains true if the Appell equation is
generalized to
φ = mFφ m−1 , F = F(x). (4.9.3)
m

Corollary. If φ m is an Appell polynomial, then φ 0 = α 0 = constant, A =
0, and, hence, A is independent of x, that is,
|φ m (x)| n = |φ m (0)| n = |α m | n , 0 ≤ m ≤ 2n − 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.

If φ m satisfies (4.9.3) and φ 0 = constant, it does not follows that φ m is
an Appell polynomial. For example, if
2 −m/2
φ m =(1 − x ) P m ,