Appell sequence
The sequence of polynomials
(1) |
with
is a geometric sequence and has trivially the properties
(2) |
and
(3) |
(see the binomial theorem). There are also other polynomial sequences (1) having these properties, for example the sequences of the Bernoulli polynomials, the Euler polynomials
and the Hermite polynomials
. Such sequences are called Appell sequences and their members are sometimes characterised as generalised monomials
, because of resemblance to the geometric sequence.
Given the first member , which must be a nonzero constant polynomial, of any Appell sequence (1), the other members are determined recursively by
(4) |
as one gives the values of the constants of integration ; thus the number sequence
determines the Appell sequence uniquely. So the choice yields a geometric sequence and the choice for the Bernoulli polynomials (http://planetmath.org/BernoulliPolynomialsAndNumbers).
The properties (2) and (3) are
equivalent (http://planetmath.org/Equivalent3). The implication
may be shown by
induction
(http://planetmath.org/Induction) on . The reverse
implication is gotten by using the definition of derivative
:
See also http://en.wikipedia.org/wiki/Appell_polynomialsWiki.
Title | Appell sequence |
Canonical name | AppellSequence |
Date of creation | 2014-05-23 17:08:17 |
Last modified on | 2014-05-23 17:08:17 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 15 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 26A99 |
Classification | msc 12-00 |
Classification | msc 11C08 |
Classification | msc 11B83 |
Classification | msc 11B68 |
Related topic | BinomialCoefficient |
Related topic | HermitePolynomials |
Related topic | HermiteNumbers |
Defines | generalized monomials |