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| In [[mathematics]], a '''Sheffer sequence''' or '''poweroid''' is a [[polynomial sequence]], i.e., a sequence { ''p''<sub>''n''</sub>(''x'') : ''n'' = 0, 1, 2, 3, . . . } 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]].
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| ==Definition==
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| Fix a polynomial sequence ''p''<sub>''n''</sub>. Define a linear operator ''Q'' on polynomials in ''x'' by
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| :<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]].
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| ==Properties==
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| The set of all Sheffer sequences is a [[group (mathematics)|group]] under the operation of '''umbral composition''' of polynomial sequences, defined as follows. Suppose { ''p''<sub>''n''</sub>(x) : ''n'' = 0, 1, 2, 3, ... } and { ''q''<sub>''n''</sub>(x) : ''n'' = 0, 1, 2, 3, ... } are polynomial sequences, given by
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| :<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>
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| Then the umbral composition <math>p \circ q</math> is the polynomial sequence whose ''n''th term is
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| :<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>
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| (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).
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| The neutral element of this group is the standard monomial basis
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| :<math>e_n(x) = x^n = \sum_{k=0}^n \delta_{n,k} x^k.</math>
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| 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
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| :<math>p_n(x+y)=\sum_{k=0}^n{n \choose k}p_k(x)p_{n-k}(y).</math> | |
| A Sheffer sequence { ''p''<sub>''n''</sub>(''x'') : ''n'' = 0, 1, 2, . . . } is of binomial type if and only if both
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| :<math>p_0(x) = 1\,</math>
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| and
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| :<math>p_n(0) = 0\mbox{ for } n \ge 1. \,</math>
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| 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 – called the "[[delta operator]]" of that sequence – 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.)
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| 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
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| :<math>s_n(x+y)=\sum_{k=0}^n{n \choose k}p_k(x)s_{n-k}(y).</math>
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| 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''<sub>''n''</sub>(''x'') } is an Appell sequence, then
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| :<math>s_n(x+y)=\sum_{k=0}^n{n \choose k}x^ks_{n-k}(y).</math> | |
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| 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.
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| A Sheffer sequence ''p''<sub>''n''</sub> is characterised by its [[exponential generating function]]
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| :<math> \sum_{n=0}^\infty \frac{p_n(x)}{n!} t^n = A(t) \exp(x B(t)) \, </math>
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| 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]].
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| ==Examples==
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| Examples of polynomial sequences which are Sheffer sequences include:
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| * The [[Abel polynomials]];
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| * The [[Bernoulli polynomials]];
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| * The central factorial polynomials;
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| * The [[Hermite polynomials]];
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| * The [[Laguerre polynomials]];
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| * The [[Mahler polynomials]];
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| * The [[monomial]]s { ''x<sup>n</sup>'' : ''n'' = 0, 1, 2, ... } ;
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| * The [[Mott polynomials]];
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| ==References==
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| * {{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.
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| * {{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 }}
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| * {{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 }}
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| *{{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}}}}
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| ==External links==
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| * {{MathWorld|ittle=Sheffer Sequence|id=ShefferSequence}}
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| [[Category:Polynomials]]
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| [[Category:Factorial and binomial topics]]
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Hi there. My name is Sophia Meagher even though it is not the name on my beginning certification. Invoicing is my occupation. What me and my family love is bungee jumping but I've been taking on new things recently. I've always cherished residing in Mississippi.
My blog post; accurate psychic readings (http://dore.gia.ncnu.edu.tw/88ipart/node/2672708)