In mathematics, a polynomial sequence
{
p
n
(
z
)
}
has a generalized Appell representation if the generating function for the polynomials takes on a certain form:
K
(
z
,
w
)
=
A
(
w
)
Ψ
(
z
g
(
w
)
)
=
∑
n
=
0
∞
p
n
(
z
)
w
n
where the generating function or kernel
K
(
z
,
w
)
is composed of the series
A
(
w
)
=
∑
n
=
0
∞
a
n
w
n
with
a
0
≠
0
and
Ψ
(
t
)
=
∑
n
=
0
∞
Ψ
n
t
n
and all
Ψ
n
≠
0
and
g
(
w
)
=
∑
n
=
1
∞
g
n
w
n
with
g
1
≠
0.
Given the above, it is not hard to show that
p
n
(
z
)
is a polynomial of degree
n
.
Boas–Buck polynomials are a slightly more general class of polynomials.
The choice of
g
(
w
)
=
w
gives the class of Brenke polynomials.
The choice of
Ψ
(
t
)
=
e
t
results in the Sheffer sequence of polynomials, which include the general difference polynomials, such as the Newton polynomials.
The combined choice of
g
(
w
)
=
w
and
Ψ
(
t
)
=
e
t
gives the Appell sequence of polynomials.
The generalized Appell polynomials have the explicit representation
p
n
(
z
)
=
∑
k
=
0
n
z
k
Ψ
k
h
k
.
The constant is
h
k
=
∑
P
a
j
0
g
j
1
g
j
2
⋯
g
j
k
where this sum extends over all partitions of
n
into
k
+
1
parts; that is, the sum extends over all
{
j
}
such that
j
0
+
j
1
+
⋯
+
j
k
=
n
.
For the Appell polynomials, this becomes the formula
p
n
(
z
)
=
∑
k
=
0
n
a
n
−
k
z
k
k
!
.
Equivalently, a necessary and sufficient condition that the kernel
K
(
z
,
w
)
can be written as
A
(
w
)
Ψ
(
z
g
(
w
)
)
with
g
1
=
1
is that
∂
K
(
z
,
w
)
∂
w
=
c
(
w
)
K
(
z
,
w
)
+
z
b
(
w
)
w
∂
K
(
z
,
w
)
∂
z
where
b
(
w
)
and
c
(
w
)
have the power series
b
(
w
)
=
w
g
(
w
)
d
d
w
g
(
w
)
=
1
+
∑
n
=
1
∞
b
n
w
n
and
c
(
w
)
=
1
A
(
w
)
d
d
w
A
(
w
)
=
∑
n
=
0
∞
c
n
w
n
.
Substituting
K
(
z
,
w
)
=
∑
n
=
0
∞
p
n
(
z
)
w
n
immediately gives the recursion relation
z
n
+
1
d
d
z
[
p
n
(
z
)
z
n
]
=
−
∑
k
=
0
n
−
1
c
n
−
k
−
1
p
k
(
z
)
−
z
∑
k
=
1
n
−
1
b
n
−
k
d
d
z
p
k
(
z
)
.
For the special case of the Brenke polynomials, one has
g
(
w
)
=
w
and thus all of the
b
n
=
0
, simplifying the recursion relation significantly.
