
Appell sequence

In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence formula_1 satisfying the identity
and in which formula_3 is a non-zero constant.
Among the most notable Appell sequences besides the trivial example formula_4 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.
Equivalent characterizations of Appell sequences.
The following conditions on polynomial sequences can easily be seen to be equivalent:
Recursion formula.
Suppose
where the last equality is taken to define the linear operator formula_16 on the space of polynomials in formula_17. Let
be the inverse operator, the coefficients formula_19 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 formula_21 as representing the Appell sequence formula_22. One can define
by using the usual power series expansion of the formula_24 and the usual definition of composition of formal power series. Then we have
In the case of Hermite polynomials, this reduces to the conventional recursion formula for that sequence.
Subgroup of the Sheffer polynomials.
The set of all Appell sequences is closed under the operation of umbral composition of polynomial sequences, defined as follows. Suppose formula_27 and formula_28 are polynomial sequences, given by
Then the umbral composition formula_30 is the polynomial sequence whose formula_31th term is
(the subscript formula_31 appears in formula_22, since this is the formula_31th term of that sequence, but not in formula_36, 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. That it is abelian can be seen by considering the fact that every Appell sequence is of the form
and that umbral composition of Appell sequences corresponds to multiplication of these formal power series in the operator formula_26.
Different convention.
Another convention followed by some authors (see "Chihara") defines this concept in a different way, conflicting with Appell's original definition, by using the identity
instead.