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In [[mathematics]], a '''Sheffer sequence''' or '''poweroid'''  is a [[polynomial sequence]], i.e., a sequence {&nbsp;''p''<sub>''n''</sub>(''x'')&nbsp;: ''n''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;.&nbsp;.&nbsp;.&nbsp;} of [[polynomial]]s 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 ''p''<sub>''n''</sub>.  Define a linear operator ''Q'' on polynomials in ''x'' by


:<math>Qp_n(x) = np_{n-1}(x)\, .</math>
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This determines ''Q'' on all polynomials. The polynomial sequence ''p''<sub>''n''</sub> 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''<sub>''a''</sub> ''g''(''x'')  is a "shift" of ''g''(''x''),  then (''Qf'')(''x'') = (''Qg'')(''x'' + ''a'');  i.e., ''Q'' commutes with every [[shift operator]]: ''T''<sub>''a''</sub>''Q'' =  ''QT''<sub>''a''</sub>.    Such a ''Q'' is a [[delta operator]].
 
==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>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>(p_n\circ q)(x)=\sum_{k=0}^n a_{n,k}q_k(x)=\sum_{0\le k \le \ell \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 neutral element of this group is the standard monomial basis
 
:<math>e_n(x) = x^n = \sum_{k=0}^n \delta_{n,k} x^k.</math>
 
Two important subgroups are the group of [[Appell sequence]]s, 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
:<math>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;.&nbsp;.&nbsp;} is of binomial type if and only if both
 
:<math>p_0(x) = 1\,</math>
 
and
 
:<math>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>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>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> \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 in ''t''.  Sheffer sequences are thus examples of [[generalized Appell polynomials]] and hence have an associated [[recurrence relation]].
 
==Examples==
Examples of polynomial sequences which are Sheffer sequences include:
* The [[Abel polynomials]];
* The [[Bernoulli polynomials]];
* The central factorial polynomials;
* The [[Hermite polynomials]];
* The [[Laguerre polynomials]];
* The [[Mahler polynomials]];
* The [[monomial]]s { ''x<sup>n</sup>'' : ''n'' = 0, 1, 2, ... } ;
* The [[Mott polynomials]];
 
==References==
 
* {{cite journal | authorlink=Gian-Carlo Rota | author=G.-C. Rota | coauthors=D. Kahaner, and [[Andrew Odlyzko|A. Odlyzko]] | title=On the foundations of combinatorial theory VIII: Finite Operator Calculus | journal=Journal of Mathematical Analysis and its Applications | volume=42 | issue=3 | date=June 1973 | pages=684–750 | doi=10.1016/0022-247X(73)90172-8 }} Reprinted in the next reference.
* {{cite book | author=G.-C. Rota | title=Finite operator calculus | coauthors=P. Doubilet, C. Greene, D. Kahaner, A. Odlyzko and R. Stanley | publisher=Academic Press | year=1975 | isbn=0-12-596650-4 }}
* {{cite journal | author=I. M. Sheffer |authorlink=Isador M. Sheffer| title=Some Properties of Polynomial Sets of Type Zero | journal=[[Duke Mathematical Journal]] | volume=5 | pages=590–622 | year=1939 | doi=10.1215/S0012-7094-39-00549-1 | issue=3 }}
*{{Cite book | last1=Roman | first1=Steven | title=The umbral calculus | url=http://books.google.com/books?id=JpHjkhFLfpgC | publisher=Academic Press Inc. [Harcourt Brace Jovanovich Publishers] | location=London | series=Pure and Applied Mathematics | isbn=978-0-12-594380-2 | mr=741185 Reprinted by Dover, 2005 | year=1984 | volume=111 | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
 
==External links==
* {{MathWorld|ittle=Sheffer Sequence|id=ShefferSequence}}
 
[[Category:Polynomials]]
[[Category:Factorial and binomial topics]]

Revision as of 20:43, 21 February 2014


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