# Appell polynomials

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 [1]. The series of Appell polynomials is defined by the formal equality

(1) |

where is a formal power series with complex coefficients , and . The Appell polynomials have an explicit expression in terms of the numbers as follows:

The condition is tantamount to saying that the degree of the polynomial is .

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

be a differential operator, generally of infinite order, defined over the algebra of complex polynomials in the variable . Then

i.e. is the image of the function under the mapping , .

The class of Appell polynomials is defined as the set of all possible systems of polynomials with generating functions of the form (1). To say that a system of polynomials (of degree ) belongs to the class amounts to saying that the relationships

are valid.

The Appell polynomials of class are sometimes defined by

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

Appell polynomials of class are used to solve equations of the form:

(2) |

The formal equality for

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

where are Appell polynomials with the generating function . 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. [1]–[6]). For a more general account of the theory of Appell polynomials of class , and a number of applications, see [6].

The class contains, as special cases, a large number of classical sequences of polynomials. Examples, up to a normalization, are the Bernoulli polynomials

etc. For many other examples, see [2] and [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

(3) |

and the Appell polynomials with the more general generating functions:

(4) |

(see, for example, [2], [3], Vol. 3). If is the inverse function to the function , then the fact that the system of polynomials belongs to the class of sequences of Appell polynomials with a generating function of type (3) is equivalent to the validity of the relations

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 on the real axis (cf. [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 [2], [7], [8].

The class , where is an integer, of Appell polynomials is defined as follows: It is the set of all systems of polynomials for each of which the (formal) representation

is valid. Here, , and , , are formal power series, the free terms of which are such that the degree of the polynomial is . To say that a sequence of polynomials of degree belongs to amounts to saying that the relations

are valid. For problems on the expansion of analytic functions in series by Appell polynomials of class , see . They are closely connected with the problem of finding analytic solutions of functional equations of the type

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

in which it is assumed that , for ; these Appell polynomials are analogues of the Jacobi polynomials. The Appell polynomials are orthogonal with the weight

(5) |

to any polynomials in two variables of degree lower than , over the domain , where is the triangle: , , ; however, they do not form a system of orthogonal functions with the weight in (see, for example, [3], Vol. 2).

#### References

[1] | P.E. Appell, Ann. Sci. École Norm. Sup. , 9 (1880) pp. 119–144 |

[2] | R.P. Boas, R.C. Buck, "Polynomial expansions of analytic functions" , Springer & Acad. Press (U.S.A. & Canada) (1958) |

[3] | H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 1–3 , McGraw-Hill (1953–1955) |

[4] | B. Wood, "Generalized Szász operators for the approximation in the complex domain" SIAM J. Appl. Math. , 17 (1969) pp. 790–801 |

[5] | G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975) |

[6] | N. Bourbaki, "Elements of mathematics. Functions of a real variable" , Addison-Wesley (1976) (Translated from French) |

[7] | J. Meixner, "Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion" J. London Math. Soc. (1) , 9 (1934) pp. 6–13 |

[8] | Ch.A. Anderson, "Some properties of Appell-like polynomials" J. Math. Anal. Appl. , 19 (1967) pp. 475–491 |

[9a] | 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 |

[9b] | Yu.A. Kaz'min, "On Appell polynomials" Math. Notes , 6 : 2 (1969) pp. 556–562 Mat. Zametki , 6 : 2 (1969) pp. 161–172 |

[10] | P.E. Appell, Arch. Math. Phys. (1) , 66 (1881) pp. 238–245 |

#### 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 [a1], p. 454, for references.

#### References

[a1] | 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 |

**How to Cite This Entry:**

Appell polynomials. Yu.A. Kaz'min (originator),

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