## Polynomial expansions of analytic functions

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. 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. The Appell polynomials of class are sometimes defined by. Appell polynomials of class are used to solve equations of the form:. The formal equality for. 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.

For a more general account of the theory of Appell polynomials of class , and a number of applications, see .

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The class contains, as special cases, a large number of classical sequences of polynomials. Examples, up to a normalization, are the Bernoulli polynomials. For many other examples, see  and  , Vol. 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. 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.

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  ,  , . 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. 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.

## Power function without multiplication

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 . They are defined by the equations:. The Appell polynomials are orthogonal with the weight. 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. Log in.

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Namespaces Page Discussion. Views View View source History. Jump to: navigation , search. 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.

## Expansion of analytic functions in q-orthogonal polynomials — Arizona State University

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### Polynomial expansions of analytic functions, by Ralph P. Boas, Jr. and R. Creighton Buck

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