In mathematics, a polynomial sequence, i.e., a sequence { p_n(x) : n = 0, 1, 2, 3, ... } of polynomials in which the index of each polynomial equals its degree, is a Sheffer sequence (from Isadore M. Sheffer) if the linear operator Q on polynomials in x defined by History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... 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. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

Qp_n(x) = np_n−1(x)

is shift-equivariant. To say that Q is shift-equivariant means that if f(x) = g(x + a) is a "shift" of g(x), then (Qf)(x) = (Qg)(x + a), i.e., Q commutes with every "shift operator".

The set of all Sheffer sequences is a group under the operation of umbral composition of polynomial sequences, defined as follows. Suppose { p_n(x) : n = 0, 1, 2, 3, ... } and { q_n(x) : n = 0, 1, 2, 3, ... } are polynomial sequences, and In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

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

Two important subgroups are the group of Appell sequences, which are those sequences for which the operator Q is differentiation, and the group of sequences of binomial type, which are those that satisfy the identity Definition In mathematics, a polynomial sequence, i. ...

A Sheffer sequence { p_n(x): n = 0, 1, 2, ... } is of binomial type if and only if both

p₀(x) = 1

and

p_n(0) = 0 for n at least 1.

The group of Appell sequences is abelian; the group of sequences of binomial type is not. The group of Appell sequences is a normal subgroup; the group of sequences of binomial type is not. The group of Sheffer sequences is a semidirect product of the group of Appell sequences and the group of sequences of binomial type. It follows that each coset of the group of Appell sequences contains exactly one sequence of binomial type. Two Sheffer sequences are in the same such coset if and only if the operator Q described above -- called the "delta operator" of that sequence -- is the same linear operator in both cases. (Generally, a delta operator is a shift-equivariant linear operator on polynomials that reduces degree by one. The term is due to F. Hildebrandt.) In mathematics, an abelian group is a commutative group, i. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g-1ng is still in N. The statement N is a normal subgroup of G is written: . Another way... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... In mathematics, a delta operator is a shift-equivariant linear operator Q on the vector space of polynomials in a variable x that reduces degrees by one. ...

If s_n(x) is a Sheffer sequence and p_n(x) is the one sequence of binomial type that shares the same delta operator, then

Sometimes the term Sheffer sequence is defined to mean a sequence that bears this relation to some sequence of binomial type. In particular, if { s_n(x) } is an Appell sequence, then

The sequence of Hermite polynomials, the sequence of Bernoulli polynomials, and the sequence { xⁿ : n = 0, 1, 2, ... } are examples of Appell sequences. In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced air MEET), are a polynomial sequence defined either by (the probabilists Hermite polynomials), or sometimes by (the physicists Hermite polynomials). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. ... 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. ...

[Lots of examples and perhaps applications should be added here.]

Some of the results above first appeared in the paper referred to below.

Reference

• 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.