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In mathematics , a polynomial sequence
{
p
n
(
z
)
}
{\displaystyle \{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
{\displaystyle K(z,w)=A(w)\Psi (zg(w))=\sum _{n=0}^{\infty }p_{n}(z)w^{n}}
where the generating function or kernel
K
(
z
,
w
)
{\displaystyle K(z,w)}
is composed of the series
A
(
w
)
=
∑
n
=
0
∞
a
n
w
n
{\displaystyle A(w)=\sum _{n=0}^{\infty }a_{n}w^{n}\quad }
with
a
0
≠
0
{\displaystyle a_{0}\neq 0}
and
Ψ
(
t
)
=
∑
n
=
0
∞
Ψ
n
t
n
{\displaystyle \Psi (t)=\sum _{n=0}^{\infty }\Psi _{n}t^{n}\quad }
and all
Ψ
n
≠
0
{\displaystyle \Psi _{n}\neq 0}
and
g
(
w
)
=
∑
n
=
1
∞
g
n
w
n
{\displaystyle g(w)=\sum _{n=1}^{\infty }g_{n}w^{n}\quad }
with
g
1
≠
0.
{\displaystyle g_{1}\neq 0.}
Given the above, it is not hard to show that
p
n
(
z
)
{\displaystyle p_{n}(z)}
is a polynomial of degree
n
{\displaystyle n}
.
Boas–Buck polynomials are a slightly more general class of polynomials.
Special cases
Explicit representation
The generalized Appell polynomials have the explicit representation
p
n
(
z
)
=
∑
k
=
0
n
z
k
Ψ
k
h
k
.
{\displaystyle p_{n}(z)=\sum _{k=0}^{n}z^{k}\Psi _{k}h_{k}.}
The constant is
h
k
=
∑
P
a
j
0
g
j
1
g
j
2
⋯
g
j
k
{\displaystyle h_{k}=\sum _{P}a_{j_{0}}g_{j_{1}}g_{j_{2}}\cdots g_{j_{k}}}
where this sum extends over all compositions of
n
{\displaystyle n}
into
k
+
1
{\displaystyle k+1}
parts; that is, the sum extends over all
{
j
}
{\displaystyle \{j\}}
such that
j
0
+
j
1
+
⋯
+
j
k
=
n
.
{\displaystyle j_{0}+j_{1}+\cdots +j_{k}=n.\,}
For the Appell polynomials, this becomes the formula
p
n
(
z
)
=
∑
k
=
0
n
a
n
−
k
z
k
k
!
.
{\displaystyle p_{n}(z)=\sum _{k=0}^{n}{\frac {a_{n-k}z^{k}}{k!}}.}
Recursion relation
Equivalently, a necessary and sufficient condition that the kernel
K
(
z
,
w
)
{\displaystyle K(z,w)}
can be written as
A
(
w
)
Ψ
(
z
g
(
w
)
)
{\displaystyle A(w)\Psi (zg(w))}
with
g
1
=
1
{\displaystyle g_{1}=1}
is that
∂
K
(
z
,
w
)
∂
w
=
c
(
w
)
K
(
z
,
w
)
+
z
b
(
w
)
w
∂
K
(
z
,
w
)
∂
z
{\displaystyle {\frac {\partial K(z,w)}{\partial w}}=c(w)K(z,w)+{\frac {zb(w)}{w}}{\frac {\partial K(z,w)}{\partial z}}}
where
b
(
w
)
{\displaystyle b(w)}
and
c
(
w
)
{\displaystyle c(w)}
have the power series
b
(
w
)
=
w
g
(
w
)
d
d
w
g
(
w
)
=
1
+
∑
n
=
1
∞
b
n
w
n
{\displaystyle b(w)={\frac {w}{g(w)}}{\frac {d}{dw}}g(w)=1+\sum _{n=1}^{\infty }b_{n}w^{n}}
and
c
(
w
)
=
1
A
(
w
)
d
d
w
A
(
w
)
=
∑
n
=
0
∞
c
n
w
n
.
{\displaystyle c(w)={\frac {1}{A(w)}}{\frac {d}{dw}}A(w)=\sum _{n=0}^{\infty }c_{n}w^{n}.}
Substituting
K
(
z
,
w
)
=
∑
n
=
0
∞
p
n
(
z
)
w
n
{\displaystyle K(z,w)=\sum _{n=0}^{\infty }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
)
.
{\displaystyle z^{n+1}{\frac {d}{dz}}\left[{\frac {p_{n}(z)}{z^{n}}}\right]=-\sum _{k=0}^{n-1}c_{n-k-1}p_{k}(z)-z\sum _{k=1}^{n-1}b_{n-k}{\frac {d}{dz}}p_{k}(z).}
For the special case of the Brenke polynomials, one has
g
(
w
)
=
w
{\displaystyle g(w)=w}
and thus all of the
b
n
=
0
{\displaystyle b_{n}=0}
, simplifying the recursion relation significantly.
See also
References
Ralph P. Boas, Jr. and R. Creighton Buck, Polynomial Expansions of Analytic Functions (Second Printing Corrected) , (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263.
Brenke, William C. (1945). "On generating functions of polynomial systems". American Mathematical Monthly . 52 (6): 297–301. doi :10.2307/2305289 .
Huff, W. N. (1947). "The type of the polynomials generated by f(xt) φ(t)". Duke Mathematical Journal . 14 (4): 1091–1104. doi :10.1215/S0012-7094-47-01483-X .