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Concept
Appell sequence
Summary
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence satisfying the identity
and in which is a non-zero constant.
Among the most notable Appell sequences besides the trivial example are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences. Appell sequences have a probabilistic interpretation as systems of moments.
The following conditions on polynomial sequences can easily be seen to be equivalent:
For ,
and is a non-zero constant;
For some sequence of scalars with ,
For the same sequence of scalars,
where
For ,
Suppose
where the last equality is taken to define the linear operator on the space of polynomials in . Let
be the inverse operator, the coefficients being those of the usual reciprocal of a formal power series, so that
In the conventions of the umbral calculus, one often treats this formal power series as representing the Appell sequence . One can define
by using the usual power series expansion of the and the usual definition of composition of formal power series. Then we have
(This formal differentiation of a power series in the differential operator is an instance of Pincherle differentiation.)
In the case of Hermite polynomials, this reduces to the conventional recursion formula for that sequence.
The set of all Appell sequences is closed under the operation of umbral composition of polynomial sequences, defined as follows. Suppose and are polynomial sequences, given by
Then the umbral composition is the polynomial sequence whose th term is
(the subscript appears in , since this is the th term of that sequence, but not in , since this refers to the sequence as a whole rather than one of its terms).
Under this operation, the set of all Sheffer sequences is a non-abelian group, but the set of all Appell sequences is an abelian subgroup.
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