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.
Definition Fix a polynomial sequence ( pn ) . Define a linear operator Q on polynomials in x by
Q [ p n ( x ) ] = n p n − 1 ( x ) . {\displaystyle Q[p_{n}(x)]=np_{n-1}(x)~.}
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 .
Properties 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
p n ( x ) = ∑ k = 0 n a n , k x k and q n ( x ) = ∑ k = 0 n b n , k x k . {\displaystyle 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}~.}
Then the umbral composition p ∘ q {\displaystyle p\circ q} is the polynomial sequence whose nth term is
( p n ∘ q ) ( x ) = ∑ k = 0 n a n , k q k ( x ) = ∑ 0 ≤ ℓ ≤ k ≤ n a n , k b k , ℓ x ℓ {\displaystyle (p_{n}\circ q)(x)=\sum _{k=0}^{n}a_{n,k}q_{k}(x)=\sum _{0\leq \ell \leq k\leq n}a_{n,k}b_{k,\ell }x^{\ell }}
(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
e n ( x ) = x n = ∑ k = 0 n δ n , k x k . {\displaystyle e_{n}(x)=x^{n}=\sum _{k=0}^{n}\delta _{n,k}x^{k}.}
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
p n ( x + y ) = ∑ k = 0 n ( n k ) p k ( x ) p n − k ( y ) . {\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}\ {n \choose k}\ p_{k}(x)\ p_{n-k}(y)~.}
A Sheffer sequence ( pn(x) : n = 0, 1, 2, ... ) is of binomial type if and only if both
p 0 ( x ) = 1 {\displaystyle p_{0}(x)=1\ }
and
p n ( 0 ) = 0 for n ≥ 1 . {\displaystyle p_{n}(0)=0\quad {\mbox{ for }}\quad n\geq 1~.}
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