Sheffer sequence

In mathematics, a Sheffer sequence or poweroid is 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, satisfying conditions related to the umbral calculus in combinatorics. They are named for Isador M. Sheffer.

Fix a polynomial sequence p_n. Define a linear operator Q on polynomials in x by

This determines Q on all polynomials. The polynomial sequence p_n 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(x) = g(x + a) = T_ag(x) is a "shift" of g(x), then (Qf)(x) = (Qg)(x + a); i.e., Q commutes with every shift operator: T_aQ =QT_a. Such a Q is a delta 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, given by

Then the umbral composition ${\displaystyle p\circ q}$ is the polynomial sequence whose nth term is

(the subscript n appears in p_n, 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).

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