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.
Equivalent characterizations of Appell sequences
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,
- For ,
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.
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 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. 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
- Appell, Paul (1880). "Sur une classe de polynômes". Annales Scientifiques de l'École Normale Supérieure. 2e Série. 9: 119–144.
- Roman, Steven; Rota, Gian-Carlo (1978). "The Umbral Calculus". Advances in Mathematics. 27 (2): 95–188. doi:10.1016/0001-8708(78)90087-7..
- Rota, Gian-Carlo; Kahaner, D.; Odlyzko, Andrew (1973). "Finite Operator Calculus". Journal of Mathematical Analysis and Applications. 42 (3): 685–760. doi:10.1016/0022-247X(73)90172-8. Reprinted in the book with the same title, Academic Press, New York, 1975.
- Steven Roman. The Umbral Calculus. Dover Publications.
- Theodore Seio Chihara (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York. ISBN 978-0-677-04150-6.