In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the area of complex analysis

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, the general difference polynomials are a polynomial sequence

Polynomial sequence

In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial...

, a certain subclass of the Sheffer polynomials, which include the Newton polynomial

Newton polynomial

In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form...

s, Selberg's polynomials, and the Stirling interpolation polynomials as special cases.

Definition

The general difference polynomial sequence is given by


where is the binomial coefficient

Binomial coefficient

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

. For , the generated polynomials are the Newton polynomials


The case of generates Selberg's polynomials, and the case of generates Stirling's interpolation polynomials.

Moving differences

Given an analytic function

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

 , define the moving difference of f as


where is the forward difference operator. Then, provided that f obeys certain summability conditions, then it may be represented in terms of these polynomials as


The conditions for summability (that is, convergence) for this sequence is a fairly complex topic; in general, one may say that a necessary condition is that the analytic function be of less than exponential type. Summability conditions are discussed in detail in Boas & Buck.

Generating function

The generating function

Generating function

In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...

 for the general difference polynomials is given by


This generating function can be brought into the form of the generalized Appell representation


by setting , , and .