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 pn. Define a linear operatorQ on polynomials in x by This determines Q on all polynomials. The polynomial sequence pn is a Sheffer sequence if the linear operator Q just defined is shift-equivariant. Here, we define a linear operator Q on polynomials to be shift-equivariant if, whenever f = g = Tag is a "shift" of g, then = ; i.e., Q commutes with every shift operator: TaQ =QTa. Such a Q is a delta 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 nth term is . The neutral 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 Q 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 sn is a Sheffer sequence and pn 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 pn is characterised by its exponential generating function where A and B are power series in t. 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:
