Appell sequence
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence {pn(x)}n = 0, 1, 2, ... satisfying the identity
and in which p0(x) is a non-zero constant.
Among the most notable Appell sequences besides the trivial example { xn } 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.
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 S on the space of polynomials in x. Let
be the inverse operator, the coefficients ak 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 T as representing the Appell sequence {pn}. One can define
by using the usual power series expansion of the log(1 + x) and the usual definition of composition of formal power series. Then we have
(This formal differentiation of a power series in the differential operator D 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 { pn(x) : n = 0, 1, 2, 3, ... } and { qn(x) : n = 0, 1, 2, 3, ... } are polynomial sequences, given by
Then the umbral composition p o q is the polynomial sequence whose nth term is
(the subscript n appears in pn, since this is the n term of that sequence, but not in q, since this refers to the sequence as a whole rather than one of its terms).
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