Editing Sheffer sequence (section)

==Properties==

The set of all Sheffer sequences is a [[group (mathematics)|group]] under the operation of '''umbral composition''' of polynomial sequences, defined as follows. Suppose (&nbsp;''p''<sub>''n''</sub>(x) : ''n'' = 0, 1, 2, 3,&nbsp;...&nbsp;) and (&nbsp;''q''<sub>''n''</sub>(x) : ''n'' = 0, 1, 2, 3,&nbsp;...&nbsp;) are polynomial sequences, given by
<math display="block">p_n(x)=\sum_{k=0}^n a_{n,k}x^k\ \mbox{and}\ q_n(x)=\sum_{k=0}^n b_{n,k}x^k.</math>

Then the umbral composition <math>p \circ q</math> is the polynomial sequence whose ''n''th term is
<math display="block">(p_n\circ q)(x) = \sum_{k=0}^n a_{n,k}q_k(x) = \sum_{0\le \ell \le k \le n} a_{n,k}b_{k,\ell}x^\ell</math>
(the subscript ''n'' appears in ''p''<sub>''n''</sub>, 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
<math display="block">e_n(x) = x^n = \sum_{k=0}^n \delta_{n,k} x^k.</math>

Two important [[subgroup]]s are the group of [[Appell sequence]]s, which are those sequences for which the operator ''Q'' is mere [[derivative|differentiation]], and the group of sequences of [[binomial type]], which are those that satisfy the identity
<math display="block">p_n(x+y) = \sum_{k=0}^n{n \choose k}p_k(x)p_{n-k}(y).</math>
A Sheffer sequence (&nbsp;''p''<sub>''n''</sub>(''x'')&nbsp;: ''n''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;...&nbsp;) is of binomial type if and only if both
<math display="block">p_0(x) = 1\,</math>
and
<math display="block">p_n(0) = 0\mbox{ for } n \ge 1. \,</math>

The group of Appell sequences is [[abelian group|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 &ndash; called the "[[delta operator]]" of that sequence &ndash; is the same linear operator in both cases. (Generally, a ''delta operator'' is a shift-equivariant linear operator on polynomials that reduces degree by one. The term is due to F. Hildebrandt.)

If ''s''<sub>''n''</sub>(''x'') is a Sheffer sequence and ''p''<sub>''n''</sub>(''x'') is the one sequence of binomial type that shares the same delta operator, then
<math display="block">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 (&nbsp;''s''<sub>''n''</sub>(''x'') ) is an Appell sequence, then
<math display="block">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 [[monomial]]s ( ''x<sup>n</sup>'' : ''n'' = 0, 1, 2, ... ) are examples of Appell sequences.

A Sheffer sequence ''p''<sub>''n''</sub> is characterised by its [[exponential generating function]]
<math display="block"> \sum_{n=0}^\infty \frac{p_n(x)}{n!} t^n = A(t) \exp(x B(t)) \, </math>
where ''A'' and ''B'' are ([[formal power series|formal]]) [[power series]] in ''t''. Sheffer sequences are thus examples of [[generalized Appell polynomials]] and hence have an associated [[recurrence relation]].