Appell sequence
Encyclopedia
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.
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.
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).
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 D.
instead.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an Appell sequence, named after Paul Émile Appell
Paul Émile Appell
Paul Appell , also known as Paul Émile Appel, was a French mathematician and Rector of the University of Paris...
, is any polynomial sequence
Polynomial sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial...
{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
Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where...
, the Bernoulli polynomials
Bernoulli polynomials
In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator...
, 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, ...,
- and p0(x) is a non-zero constant;
- For some sequence {cn}n = 0, 1, 2, ... of scalars with c0 ≠ 0,
- For the same sequence of scalars,
- where
- For n = 0, 1, 2, ...,
Recursion formula
Supposewhere 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
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
, so that
In the conventions of the umbral calculus
Umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to 'prove' them. These techniques were introduced by and are sometimes called Blissard's symbolic method...
, 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
Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where...
, 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 byThen 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).
Under this operation, the set of all Sheffer sequences is a non-abelian group, but the set of all Appell sequences is an abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
. 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
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
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 identityinstead.
See also
- Sheffer sequence
- Umbral calculusUmbral calculusIn mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to 'prove' them. These techniques were introduced by and are sometimes called Blissard's symbolic method...
- Generalized Appell polynomials
- Wick productWick productIn probability theory, the Wick product\langle X_1,\dots,X_k \rangle\,named after physicist Gian-Carlo Wick, is a sort of product of the random variables, X1, ..., Xk, defined recursively as follows:\langle \rangle = 1\,...
External links
- Appell Sequence at MathWorldMathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...