Sheffer sequence


In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are named for Isador M. Sheffer.

Definition

Fix a polynomial sequence Define a linear operator on polynomials in by
This determines on all polynomials. The polynomial sequence is a Sheffer sequence if the linear operator just defined is shift-equivariant; such a is then a delta operator. Here, we define a linear operator on polynomials to be shift-equivariant if, whenever is a "shift" of then i.e., commutes with every shift operator:.

Properties

The set of all Sheffer sequences is a group 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 identity element of this group is the standard monomial basis
Two important subgroups are the group of Appell sequences, which are those sequences for which the operator is mere differentiation, and the group of sequences of binomial type, which are those that satisfy the identity
A Sheffer sequence is of binomial type if and only if both
and
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.
If is a Sheffer sequence and 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 is an Appell sequence, then
The sequence of Hermite polynomials, the sequence of Bernoulli polynomials, and the monomials are examples of Appell sequences.
A Sheffer sequence is characterised by its exponential generating function
where and are power series in. Sheffer sequences are thus examples of generalized Appell polynomials and hence have an associated recurrence relation.

Examples

Examples of polynomial sequences which are Sheffer sequences include: