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
Appell sequences have a probabilistic interpretation as systems of moments.
The following conditions on polynomial sequences can easily be seen to be equivalent: 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.
are polynomial sequences, given by Then the umbral composition
is the polynomial sequence whose
th term is (the subscript
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.
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
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.
The enormous class of Appell polynomials can be obtained in terms of the generalized hypergeometric function.
denote the array of
ratios Consider the polynomial
is the generalized hypergeometric function.
The polynomial family
is the Appell sequence for any natural parameters