Inmathematics, aSheffer sequence orpoweroid is apolynomial sequence, i.e., asequence(p_n(x) :n = 0, 1, 2, 3, ...) ofpolynomials in which the index of each polynomial equals itsdegree, satisfying conditions related to theumbral calculus incombinatorics. They are named forIsador M. Sheffer.

Fix a polynomial sequence (p_n). Define alinear operatorQ on polynomials inx by$${\displaystyle Q{p}_n(x)=n{p}_{n-1}(x).}$$ 

This determinesQ on all polynomials. The polynomial sequencep_n is aSheffer sequence if the linear operatorQ just defined isshift-equivariant; such aQ is then adelta operator. Here, we define a linear operatorQ on polynomials to beshift-equivariant if, wheneverf(x) =g(x +a) =T_ag(x) is a "shift" ofg(x), then (Qf)(x) = (Qg)(x +a); i.e.,Q commutes with everyshift operator:T_aQ =QT_a.

The set of all Sheffer sequences is agroup under the operation ofumbral 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$${\displaystyle p_n(x)=\sum _{k=0}^n{a}_{n,k}{x}^k\text{and}{q}_n(x)=\sum _{k=0}^n{b}_{n,k}{x}^k.}$$ 

Then the umbral composition${\displaystyle p\circ q}$  is the polynomial sequence whosenth term is$${\displaystyle (p_n\circ q)(x)=\sum _{k=0}^n{a}_{n,k}{q}_k(x)=\sum _{0\le \ell \le k\le n}{a}_{n,k}{b}_{k,\ell }{x}^{\ell }}$$ (the subscriptn appears inp_n, since this is then term of that sequence, but not inq, 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$${\displaystyle e_n(x)=x^n=\sum _{k=0}^n{\delta }_{n,k}{x}^k.}$$ 

Two importantsubgroups are the group ofAppell sequences, which are those sequences for which the operatorQ is meredifferentiation, and the group of sequences ofbinomial type, which are those that satisfy the identity$${\displaystyle p_n(x+y)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})p_k(x)p_{n-k}(y).}$$ A Sheffer sequence ( p_n(x) :n = 0, 1, 2, ... ) is of binomial type if and only if both$${\displaystyle p_0(x)=1}$$ and$${\displaystyle p_n(0)=0\text{ for }n\ge 1.}$$ 

The group of Appell sequences isabelian; the group of sequences of binomial type is not. The group of Appell sequences is anormal subgroup; the group of sequences of binomial type is not. The group of Sheffer sequences is asemidirect product of the group of Appell sequences and the group of sequences of binomial type. It follows that eachcoset 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 operatorQ described above – called the "delta operator" of that sequence – is the same linear operator in both cases. (Generally, adelta operator is a shift-equivariant linear operator on polynomials that reduces degree by one. The term is due to F. Hildebrandt.)

Ifs_n(x) is a Sheffer sequence andp_n(x) is the one sequence of binomial type that shares the same delta operator, then$${\displaystyle s_n(x+y)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})p_k(x)s_{n-k}(y).}$$ 

Sometimes the termSheffer sequence isdefined 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$${\displaystyle s_n(x+y)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^k{s}_{n-k}(y).}$$ 

The sequence ofHermite polynomials, the sequence ofBernoulli polynomials, and themonomials (xⁿ :n = 0, 1, 2, ... ) are examples of Appell sequences.

A Sheffer sequencep_n is characterised by itsexponential generating function$${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{p_n(x)}{n!}{t}^n=A(t)\exp (xB(t))}$$ whereA andB are (formal)power series int. Sheffer sequences are thus examples ofgeneralized Appell polynomials and hence have an associatedrecurrence relation.

Examples of polynomial sequences which are Sheffer sequences include:

• TheAbel polynomials;
• TheBernoulli polynomials;
• TheEuler polynomial;
• The central factorial polynomials;
• TheHermite polynomials;
• TheLaguerre polynomials;
• Themonomials (xⁿ :n = 0, 1, 2, ... );
• TheMott polynomials;
• TheBernoulli polynomials of the second kind;
• TheFalling and rising factorials;
• TheTouchard polynomials;
• TheMittag-Leffler polynomials;

• Weisstein, Eric W."Sheffer Sequence".MathWorld.