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

\( {d \over dx} p_n(x) = np_{n-1}(x), \)

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.

Equivalent characterizations of Appell sequences

The following conditions on polynomial sequences can easily be seen to be equivalent:

For n = 1, 2, 3, ...,

\( {d \over dx} p_n(x) = np_{n-1}(x) \)

and p0(x) is a non-zero constant;

For some sequence {cn}n = 0, 1, 2, ... of scalars with c0 ≠ 0,

\( p_n(x) = \sum_{k=0}^n {n \choose k} c_k x^{n-k}; \)

For the same sequence of scalars,

\( p_n(x) = \left(\sum_{k=0}^\infty {c_k \over k!} D^k\right) x^n, \)

where

D = {d \over dx};

For n = 0, 1, 2, ...,

p_n(x+y) = \sum_{k=0}^n {n \choose k} p_k(x) y^{n-k}.

Recursion formula

Suppose

\( p_n(x) = \left(\sum_{k=0}^\infty {c_k \over k!} D^k\right) x^n = Sx^n, \)

where the last equality is taken to define the linear operator S on the space of polynomials in x. Let

\( T = S^{-1} = \left(\sum_{k=0}^\infty {c_k \over k!} D^k\right)^{-1} = \sum_{k=1}^\infty {a_k \over k!} D^k \)

be the inverse operator, the coefficients ak being those of the usual reciprocal of a formal power series, so that

\( Tp_n(x) = x^n.\, \)

In the conventions of the umbral calculus, one often treats this formal power series T as representing the Appell sequence {pn}. One can define

\( \log T = \log\left(\sum_{k=0}^\infty {a_k \over k!} D^k \right) \)

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

\( p_{n+1}(x) = (x - (\log T)')p_n(x).\, \)

(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.
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 { pn(x) : n = 0, 1, 2, 3, ... } and { qn(x) : n = 0, 1, 2, 3, ... } are polynomial sequences, given by

\( p_n(x)=\sum_{k=0}^n a_{n,k}x^k\ \mbox{and}\ q_n(x)=\sum_{k=0}^n b_{n,k}x^k. \)

Then the umbral composition p o q is the polynomial sequence whose nth term is

\( (p_n\circ q)(x)=\sum_{k=0}^n a_{n,k}q_k(x)=\sum_{0\le k \le \ell \le n} a_{n,k}b_{k,\ell}x^\ell \)

(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).

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

\( p_n(x) = \left(\sum_{k=0}^\infty {c_k \over k!} D^k\right) x^n, \)

and that umbral composition of Appell sequences corresponds to multiplication of these formal power series in the operator D.
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

\( {d \over dx} p_n(x) = p_{n-1}(x) \)

instead.
See also

Sheffer sequence
Umbral calculus
Generalized Appell polynomials
Wick product

References

Paul Appell, "Sur une classe de polynômes", Annales scientifiques de l'École Normale Supérieure 2e série, tome 9, 1880.
Steven Roman and Gian-Carlo Rota, "The Umbral Calculus", Advances in Mathematics, volume 27, pages 95 – 188, (1978).
G.-C. Rota, D. Kahaner, and A. Odlyzko, "Finite Operator Calculus", Journal of Mathematical Analysis and its Applications, vol. 42, no. 3, June 1973. 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 0-677-04150-0.

External links

Appell Sequence at MathWorld

Mathematics Encyclopedia