Appell sequence
type of polynomial sequence
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence
{
p
n
(
x
)
}
n
=
0
,
1
,
2
,
…
{\displaystyle \{p_{n}(x)\}_{n=0,1,2,\ldots }}
satisfying the identity
d
d
x
p
n
(
x
)
=
n
p
n
−
1
(
x
)
,
{\displaystyle {\frac {d}{dx}}p_{n}(x)=np_{n-1}(x),}
and in which
p
0
(
x
)
{\displaystyle p_{0}(x)}
is a non-zero constant.
Among the most notable Appell sequences besides the trivial example
{
x
n
}
{\displaystyle \{x^{n}\}}
are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences. Appell sequences have a probabilistic interpretation as systems of moments.
Equivalent characterizations of Appell sequences
The following conditions on polynomial sequences can easily be seen to be equivalent:
For
n
=
1
,
2
,
3
,
…
{\displaystyle n=1,2,3,\ldots }
,
d
d
x
p
n
(
x
)
=
n
p
n
−
1
(
x
)
{\displaystyle {\frac {d}{dx}}p_{n}(x)=np_{n-1}(x)}
and
p
0
(
x
)
{\displaystyle p_{0}(x)}
is a non-zero constant;
For some sequence
{
c
n
}
n
=
0
∞
{\textstyle \{c_{n}\}_{n=0}^{\infty }}
of scalars with
c
0
≠
0
{\displaystyle c_{0}\neq 0}
,
p
n
(
x
)
=
∑
k
=
0
n
(
n
k
)
c
k
x
n
−
k
;
{\displaystyle p_{n}(x)=\sum _{k=0}^{n}{\binom {n}{k}}c_{k}x^{n-k};}
For the same sequence of scalars,
p
n
(
x
)
=
(
∑
k
=
0
∞
c
k
k
!
D
k
)
x
n
,
{\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)x^{n},}
where
D
=
d
d
x
;
{\displaystyle D={\frac {d}{dx}};}
For
n
=
0
,
1
,
2
,
…
{\displaystyle n=0,1,2,\ldots }
,
p
n
(
x
+
y
)
=
∑
k
=
0
n
(
n
k
)
p
k
(
x
)
y
n
−
k
.
{\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}p_{k}(x)y^{n-k}.}
Recursion formula
Suppose
p
n
(
x
)
=
(
∑
k
=
0
∞
c
k
k
!
D
k
)
x
n
=
S
x
n
,
{\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{c_{k} \over k!}D^{k}\right)x^{n}=Sx^{n},}
where the last equality is taken to define the linear operator
S
{\displaystyle S}
on the space of polynomials in
x
{\displaystyle x}
. Let
T
=
S
−
1
=
(
∑
k
=
0
∞
c
k
k
!
D
k
)
−
1
=
∑
k
=
1
∞
a
k
k
!
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