Sheffer sequence

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 if the the linear operator Q on polynomials in x defined by

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

<math>p_n(x)=\sum_{k=0}^n a_{n,k}x^k.</math>

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

<math>(p_n\circ q)(x)=\sum_{k=0}^n a_{n,k}q_k(x).</math>

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

<math>p_n(x+y)=\sum_{k=0}^n{n \choose k}p_k(x)p_{n-k}(y).</math>

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 is a shift-equivariant linear operator on polynomials that reduces degree by one. The term is due to F. Hildebrandt.)

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

<math>s_n(x+y)=\sum_{k=0}^n{n \choose k}p_k(x)s_{n-k}(y).</math>

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

<math>s_n(x+y)=\sum_{k=0}^n{n \choose k}x^ks_{n-k}(y).</math>

The sequence of Hermite polynomials, the sequence of Bernoulli polynomials, and the sequence { xⁿ : n = 0, 1, 2, ... } are examples of Appell sequences.

[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.