In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the area of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the general difference polynomials are a
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. Polynomial sequences are a topic of interest in en ...
, a certain subclass of the
Sheffer polynomials, which include the
Newton polynomials, 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
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
. 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
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. Functions of each type are infinitely differentiable, but complex an ...
, 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
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
for the general difference polynomials is given by
:
This generating function can be brought into the form of the
generalized Appell representation In mathematics, a polynomial sequence \ has a generalized Appell representation if the generating function for the polynomials takes on a certain form:
:K(z,w) = A(w)\Psi(zg(w)) = \sum_^\infty p_n(z) w^n
where the generating function or kernel K(z ...
:
by setting
,
,
and
.
See also
*
Carlson's theorem
In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions which do not grow very fast at infinity can not co ...
*
Bernoulli polynomials of the second kind
The Bernoulli polynomials of the second kind , also known as the Fontana-Bessel polynomials, are the polynomials defined by the following generating function:
:
\frac= \sum_^\infty z^n \psi_n(x) ,\qquad , z, -1
and
:\gamma=\sum_^\infty\frac\B ...
References
{{reflist
*
Ralph P. Boas, Jr.
Ralph Philip Boas Jr. (August 8, 1912 – July 25, 1992) was a mathematician, teacher, and journal editor. He wrote over 200 papers, mainly in the fields of real analysis, real and complex analysis..
Biography
He was born in Walla Walla, Washi ...
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.
Polynomials
Finite differences
Factorial and binomial topics