Page 332 - Determinants and Their Applications in Mathematical Physics

A.4 Appell Polynomials 317

Appell Sets
Any sequence of polynomials {φ m (x)} where φ m (x) is of exact degree m
and satisfies the Appell equation (A.4.5) is known as an Appell set.
The sequence in which
−1

m + s
φ m (x)= (x + c) m+s , s =1, 2, 3,...,
s
satisfies (A.4.5), but its members are not of degree m. The sequence in
which
2 2m+1 m!(m + 1)! (x + c) m+(1/2)
φ m (x)=
(2m + 2)!
satisfies (A.4.5), but its members are not polynomials. Hence, neither
sequence is an Appell set.
Carlson proved that if {φ m } and {ψ m } are each Appell sets and
m
m
θ m =2 −m φ r ψ m−r ,
r
r=0
then {θ m } is also an Appell set.
In a paper on determinants with hypergeometric elements, Burchnall
proved that if {φ m } and {ψ m } are each Appell sets and

n
n

θ m = (−1) r φ m+n−r ψ r , n =0, 1, 2,...,
r
r=0
then {θ m } is also an Appell set for each value of n. Burchnall’s formula can
be expressed in the form

n

θ m = (−1) r ψ r φ m+n−r , n =0, 1, 2,... .
ψ r+1 φ m+n−r+1

r=0
The generalized Appell equation

θ = mf θ m−1 , f = f(x),
m
is satisfied by
θ m = φ m (f),
where φ m (x) is any solution of (A.4.5). For example, the equation
mθ m−1
θ =
(1 + x)
m 2
is satisfied by
1

θ m = φ m − .
1+ x