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Concept
Sheffer sequence
Summary
In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence (pn(x) : n = 0, 1, 2, 3, ...) 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.
Fix a polynomial sequence (pn). Define a linear operator Q 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; such a Q is then a delta operator. Here, we define a linear operator Q on polynomials to be shift-equivariant if, whenever f(x) = g(x + a) = Ta g(x) is a "shift" of g(x), then (Qf)(x) = (Qg)(x + a); i.e., Q commutes with every shift operator: TaQ = QTa.
The set of all Sheffer sequences is a group 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
Then the umbral composition 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).
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 Q is mere differentiation, and the group of sequences of binomial type, which are those that satisfy the identity
A Sheffer sequence ( pn(x) : n = 0, 1, 2, ... ) 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.
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