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A class of polynomials over the field of complex numbers which contains many classical polynomial systems. The Appell polynomials were introduced by P.E. Appell [[#References|[1]]]. The series of Appell polynomials  $  \{ A _ {n} (z) \} _ {n=0 }   ^  \infty  $
is defined by the formal equality

$$ \tag{1 }
A (t) e  ^ {zt}  = \ 
\sum _ {n = 0 } ^  \infty  
A _ {n} (z) t  ^ {n} ,
$$

where  $  A(t) = \sum _ {k=0 }   ^  \infty  a _ {k} t  ^ {k} $
is a formal power series with complex coefficients  $  a _ {k} $, 
$  k = 0, 1 \dots $
and  $  a _ {0} \neq 0 $. 
The Appell polynomials  $  {A _ {n} } (z) $
have an explicit expression in terms of the numbers  $  a _ {k} $
as follows:

$$ 
A _ {n} (z)  = \ 
\sum _ {k = 0 } ^  \infty  

\frac{a _ k}{( n - k )! }
 z ^ {n - k } ,\ \ 
n = 0, 1 ,\dots .
$$

The condition  $  a _ {0} \neq 0 $
is tantamount to saying that the degree of the polynomial  $  {A _ {n} } (z) $
is  $  n $.

There is another, equivalent, definition of Appell polynomials. Let

$$ 
A (D)  =  \sum _ {k = 0 } ^  \infty  
a _ {k} D  ^ {k} ,\ \ 
D  =  
\frac{d}{dz}
 ,\ \ 
a _ {0}  \neq  0 ,
$$

be a differential operator, generally of infinite order, defined over the algebra  $  P $
of complex polynomials in the variable  $  z = x + iy $. 
Then

$$ 
A _ {n} (z)  =  
\frac{A (D) z  ^ {n} }{n!}
 ,\ \ 
n = 0, 1 \dots
$$

i.e.  $  {A _ {n} } (z) $
is the image of the function  $  {z  ^ {n} } /n! $
under the mapping  $  p = A(D) q $, 
$  p, q \in P $.

The class  $  A  ^ {(1)} $
of Appell polynomials is defined as the set of all possible systems of polynomials  $  \{ A _ {n} (z) \} $
with generating functions of the form (1). To say that a system  $  \{ P _ {n} (z) \} $
of polynomials (of degree  $  n $) 
belongs to the class  $  A  ^ {(1)} $
amounts to saying that the relationships

$$ 
P _ {n} ^ { \prime } (z)  =  P _ {n -1 }  (z),\ \ 
n = 1, 2 \dots
$$

are valid.

The Appell polynomials of class  $  A  ^ {(1)} $
are sometimes defined by

$$ 
A (t) e  ^ {zt}  =  \sum _ {n = 0 } ^  \infty  

\frac{\widehat{A}  _ {n} (z) }{n!}
 t  ^ {n} ,
$$

$$ 
\widehat{A}  _ {n} ^ { \prime } (z)  =  n \widehat{A}  _ {n - 1 }  (z),\  n = 1, 2 \dots
$$

which, apart from normalization, are equivalent to those given above.

Appell polynomials of class  $  A  ^ {(1)} $
are used to solve equations of the form:

$$ \tag{2 }
A (D) y (z)  =  f (z).
$$

The formal equality  $  y(z) = f(z) / A(D) $
for

$$ 
f (z)  =  \sum _ {k = 0 } ^  \infty  

\frac{c _ {k} z  ^ {k} }{k!}

$$

makes it possible to write the solution of (2) in the form

$$ 
y (z)  =  \sum _ {k = 0 } ^  \infty  
c _ {k} A _ {k}  ^ {*} (z),
$$

where  $  \{ A _ {k}  ^ {*} (z) \} $
are Appell polynomials with the generating function  $  {e  ^ {zt} } / A(t) $. 
In this connection the expansion of analytic functions into Appell polynomials is of special interest. Appell polynomials also find use in various problems connected with functional equations, including differential equations other than (2), in interpolation problems, in approximation theory, in summation methods, etc. (cf. [[#References|[1]]]–[[#References|[6]]]). For a more general account of the theory of Appell polynomials of class  $  A  ^ {(1)} $, 
and a number of applications, see [[#References|[6]]].

The class  $  A  ^ {(1)} $
contains, as special cases, a large number of classical sequences of polynomials. Examples, up to a normalization, are the [[Bernoulli polynomials|Bernoulli polynomials]]

$$ 

\frac{te  ^ {zt} }{e  ^ {t} -1 }
  = \ 
\sum _ {n = 0 } ^  \infty  

\frac{B _ {n} (z) }{n!}
 t  ^ {n} ;
$$

the [[Hermite polynomials|Hermite polynomials]]

$$ 
e ^ {z t - t  ^ {2} /2 }  = \ 
\sum _ {n = 0 } ^  \infty  

\frac{H _ {n} (z) }{n!}
 t  ^ {n} ;
$$

the [[Laguerre polynomials|Laguerre polynomials]]

$$ 
( 1 - t )  ^  \alpha  e  ^ {zt}  = \ 
\sum _ {n = 0 } ^  \infty  
( - 1)  ^ {n}
L _ {n} ^ {( \alpha - n ) } (z) t  ^ {n} ;
$$

etc. For many other examples, see [[#References|[2]]] and [[#References|[3]]], Vol. 3.

There exist various generalizations of Appell polynomials, which are also known as systems of Appell polynomials. These include the Appell polynomials with generating functions of the form

$$ \tag{3 }
\left .

\begin{array}{c}
A (w) e  ^ {zU(w)}  = \ 
\sum _ {n = 0 } ^  \infty  
p _ {n} (z) w  ^ {n} ,  \\
U (w)  =  \sum _ {k = 1 } ^  \infty  
c _ {k} w  ^ {k} ,\ \ 
c _ {1} \neq 0 ,  \\
\end{array}

\right \}
$$

and the Appell polynomials with the more general generating functions:

$$ \tag{4 }
A (w) \psi ( z U (w) )  = \ 
\sum _ {n = 0 } ^  \infty  
q _ {n} (z) w  ^ {n}
$$

(see, for example, [[#References|[2]]], [[#References|[3]]], Vol. 3). If  $  w(U) $
is the inverse function to the function  $  U(w) $, 
then the fact that the system of polynomials  $  \{ p _ {n} (z) \} _ {n=0}  ^  \infty  $
belongs to the class of sequences of Appell polynomials with a generating function of type (3) is equivalent to the validity of the relations

$$ 
w (D) p _ {n} (z)  =  p _ {n-1} (z),
\  n = 1 , 2 ,\dots \  \left ( D = 
\frac{d}{dz}
 \right ) .
$$

There are only five weighted orthogonal systems of sequences of Appell polynomials on the real axis with generating functions of the type (3); these include only one orthogonal system with generating functions of the type (1), which consists of Hermite polynomials with the weight  $  e ^ {-x  ^ {2} / 2 } $
on the real axis (cf. [[#References|[7]]]).

For the expansion in series by Appell polynomials with generating functions of the types (3) and (4), and interconnections of these polynomials by various functional equations see [[#References|[2]]], [[#References|[7]]], [[#References|[8]]].

The class  $  A  ^ {(p)} $, 
where  $  p \geq  1 $
is an integer, of Appell polynomials is defined as follows: It is the set of all systems of polynomials  $  \{ A _ {n} (z) \} $
for each of which the (formal) representation

$$ 
\sum _ {k = 0 } ^ { p-1 } 
A _ {k} (t) e ^ {z t \omega _ {p}  ^ {k} }  = \ 
\sum _ {n = 0 } ^  \infty  
A _ {n} (z) t  ^ {n}
$$

is valid. Here,  $  \omega _ {p} = e ^ {2 \pi i / p } $, 
and  $  {A _ {k} } (t) $, 
$  k = 0 \dots p - 1 $, 
are formal power series, the free terms of which are such that the degree of the polynomial  $  {A _ {n} } (z) $
is  $  n $. 
To say that a sequence  $  \{ {Q _ {n} } (z) \} $
of polynomials of degree  $  n $
belongs to  $  A  ^ {(p)} $
amounts to saying that the relations

$$ 
D  ^ {p} Q _ {n} (z)  =  Q _ {n}  ^ {(p)}
(z)  =  Q _ {n-p} (z),
\  n = p , p + 1 \dots
$$

are valid. For problems on the expansion of analytic functions in series by Appell polynomials of class  $  A  ^ {(p)} $, 
see . They are closely connected with the problem of finding analytic solutions of functional equations of the type

$$ 
\sum _ {k = 0 } ^ { {p }  - 1 }
A _ {k} (D) y ( z \omega _ {p}  ^ {k} )  =  f (z).
$$

Appell polynomials in two variables were introduced by P. Appell [[#References|[10]]]. They are defined by the equations:

$$ 
J _ {m,n} ( \alpha , \gamma , \gamma  ^  \prime  , x, y )  = \ 

\frac{x ^ {1- \gamma } y ^ {1- \gamma  ^  \prime  } }{( \gamma ) _ {m} ( \gamma  ^  \prime  ) _ {n} }

( 1 - x - y ) ^ {\gamma + \gamma  ^  \prime  - \alpha } \times
$$

$$ 
\times 

\frac{\partial   ^ {m+n} }{\partial  x  ^ {m} \partial  y  ^ {n} }
 [ x ^ {
\gamma + m - 1 } y ^ {\gamma  ^  \prime  +n-1 } ( 1 -
x-y ) ^ {\alpha + m + n - \gamma - \gamma  ^  \prime  } ],
$$

$$ 
m , n = 0 , 1 \dots
$$

in which it is assumed that  $  ( \gamma ) _ {0} = 1 $, 
$  ( \gamma ) _ {n} = \gamma ( \gamma + 1 ) \dots ( \gamma + n - 1 ) $
for  $  n \geq  1 $; 
these Appell polynomials are analogues of the [[Jacobi polynomials|Jacobi polynomials]]. The Appell polynomials  $  J _ {m,n }  $
are orthogonal with the weight

$$ \tag{5 }
t ( x , y )  =  x ^ {\gamma -1 } y ^ {\gamma  ^  \prime  -1 }
( 1 - x - y ) ^ {\alpha - \gamma - \gamma  ^  \prime  }
$$

to any polynomials in two variables of degree lower than  $  m + n $, 
over the domain  $  T $, 
where  $  T $
is the triangle:  $  x > 0 $, 
$  y > 0 $, 
$  x + y < 1 $; 
however, they do not form a system of orthogonal functions with the weight  $  t(x, y) $
in  $  T $(
see, for example, [[#References|[3]]], Vol. 2).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  P.E. Appell, "Sur une classe de polynômes",  ''Ann. Sci. École Norm. Sup.'' , '''9'''  (1880)  pp. 119–144. {{ZBL|12.0342.02}}</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  R.P. Boas,   R.C. Buck,   "Polynomial expansions of analytic functions" , Springer &amp; Acad. Press (U.S.A. &amp; Canada)  (1958)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1–3''' , McGraw-Hill  (1953–1955)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  B. Wood,   "Generalized Szász operators for the approximation in the complex domain"  ''SIAM J. Appl. Math.'' , '''17'''  (1969)  pp. 790–801</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  G. Szegö,   "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Functions of a real variable" , Addison-Wesley  (1976)  (Translated from French)</TD></TR>
<TR><TD valign="top">[7]</TD> <TD valign="top">  J. Meixner,   "Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion"  ''J. London Math. Soc. (1)'' , '''9'''  (1934)  pp. 6–13</TD></TR>
<TR><TD valign="top">[8]</TD> <TD valign="top">  Ch.A. Anderson,   "Some properties of Appell-like polynomials"  ''J. Math. Anal. Appl.'' , '''19'''  (1967)  pp. 475–491</TD></TR>
<TR><TD valign="top">[9a]</TD> <TD valign="top">  Yu.A. Kaz'min,   "Expansions in series of Appell polynomials"  ''Math. Notes'' , '''5''' :  5  (1969)  pp. 304–311  ''Mat. Zametki'' , '''5''' :  5  (1969)  pp. 509–520</TD></TR>
<TR><TD valign="top">[9b]</TD> <TD valign="top">  Yu.A. Kaz'min,   "On Appell polynomials"  ''Math. Notes'' , '''6''' :  2  (1969)  pp. 556–562  ''Mat. Zametki'' , '''6''' :  2  (1969)  pp. 161–172</TD></TR>
<TR><TD valign="top">[10]</TD> <TD valign="top">  P.E. Appell, "Sur des polynômes de deux variables analogues aux polynômes de Jacobi",  ''Arch. Math. Phys. (1)'' , '''66'''  (1881)  pp. 238–245. {{ZBL|13.0389.02}}</TD></TR>
</table>

====Comments====
For Appell polynomials in two variables an explicit biorthogonal system is known. There is also an explicit system for the weight function (5), consisting of products of two Jacobi polynomials and a power. See [[#References|[a1]]], p. 454, for references.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.H. Koornwinder,   "Two-variable analogues of the classical orthogonal polynomials"  R.A. Askey (ed.) , ''Theory and applications of special functions'' , Acad. Press  (1975)  pp. 435–495</TD></TR></table>