(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
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.
|Date of creation||2014-05-23 17:08:17|
|Last modified on||2014-05-23 17:08:17|
|Last modified by||pahio (2872)|