|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[mathematics]], an '''Appell sequence''', named after [[Paul Émile Appell]], is any [[polynomial sequence]] {''p''<sub>''n''</sub>(''x'')}<sub>''n'' = 0, 1, 2, ...</sub> satisfying the identity
| | Hello! <br>I'm Korean male ;=). <br>I really like Breaking Bad!<br><br>My website: [http://wiki.Fonrestorff.pl/index.php/User:NicholaKleiber Fifa 15 Coin Generator] |
| | |
| :<math>{d \over dx} p_n(x) = np_{n-1}(x),</math>
| |
| | |
| and in which ''p''<sub>0</sub>(''x'') is a non-zero constant.
| |
| | |
| Among the most notable Appell sequences besides the trivial example { ''x''<sup>''n''</sup> } are the [[Hermite polynomials]], the [[Bernoulli polynomials]], and the [[Euler polynomials]]. Every Appell sequence is a [[Sheffer sequence]], but most Sheffer sequences are not Appell sequences.
| |
| | |
| ==Equivalent characterizations of Appell sequences==
| |
| | |
| | |
| The following conditions on polynomial sequences can easily be seen to be equivalent:
| |
| | |
| * For ''n'' = 1, 2, 3, ...,
| |
| | |
| ::<math>{d \over dx} p_n(x) = np_{n-1}(x)</math>
| |
| | |
| :and ''p''<sub>0</sub>(''x'') is a non-zero constant;
| |
| | |
| * For some sequence {''c''<sub>''n''</sub>}<sub>''n'' = 0, 1, 2, ...</sub> of scalars with ''c''<sub>0</sub> ≠ 0,
| |
| | |
| ::<math>p_n(x) = \sum_{k=0}^n {n \choose k} c_k x^{n-k};</math>
| |
| | |
| * For the same sequence of scalars,
| |
| | |
| ::<math>p_n(x) = \left(\sum_{k=0}^\infty {c_k \over k!} D^k\right) x^n,</math>
| |
| | |
| :where
| |
| | |
| ::<math>D = {d \over dx};</math>
| |
| | |
| * For ''n'' = 0, 1, 2, ...,
| |
| | |
| ::<math>p_n(x+y) = \sum_{k=0}^n {n \choose k} p_k(x) y^{n-k}.</math> | |
| | |
| ==Recursion formula==
| |
| | |
| Suppose
| |
| | |
| :<math>p_n(x) = \left(\sum_{k=0}^\infty {c_k \over k!} D^k\right) x^n = Sx^n,</math>
| |
| | |
| where the last equality is taken to define the linear operator ''S'' on the space of polynomials in ''x''. Let
| |
| | |
| :<math>T = S^{-1} = \left(\sum_{k=0}^\infty {c_k \over k!} D^k\right)^{-1} = \sum_{k=1}^\infty {a_k \over k!} D^k</math>
| |
| | |
| be the inverse operator, the coefficients ''a''<sub>''k''</sub> being those of the usual reciprocal of a [[formal power series]], so that
| |
| | |
| :<math>Tp_n(x) = x^n.\,</math>
| |
| | |
| In the conventions of the [[umbral calculus]], one often treats this formal power series ''T'' as representing the Appell sequence {''p''<sub>''n''</sub>}. One can define
| |
| | |
| :<math>\log T = \log\left(\sum_{k=0}^\infty {a_k \over k!} D^k \right) </math>
| |
| | |
| by using the usual power series expansion of the log(1 + ''x'') and the usual definition of composition of formal power series. Then we have
| |
| | |
| :<math>p_{n+1}(x) = (x - (\log T)')p_n(x).\,</math>
| |
| | |
| (This formal differentiation of a power series in the differential operator ''D'' is an instance of [[Pincherle derivative|Pincherle differentiation]].)
| |
| | |
| In the case of [[Hermite polynomials]], this reduces to the conventional recursion formula for that sequence.
| |
| | |
| ==Subgroup of the Sheffer polynomials==
| |
| | |
| The set of all Appell sequences is closed 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
| |
| | |
| :<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 ''p'' o ''q'' 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).
| |
| | |
| Under this operation, the set of all Sheffer sequences is a [[non-abelian group]], but the set of all Appell sequences is an [[abelian group|abelian]] [[subgroup]]. That it is abelian can be seen by considering the fact that every Appell sequence is of the form
| |
| | |
| :<math>p_n(x) = \left(\sum_{k=0}^\infty {c_k \over k!} D^k\right) x^n,</math>
| |
| | |
| and that umbral composition of Appell sequences corresponds to multiplication of these [[formal power series]] in the operator ''D''.
| |
| | |
| ==Different convention==
| |
| | |
| Another convention followed by some authors (see ''Chihara'') defines this concept in a different way, conflicting with Appell's original definition, by using the identity
| |
| | |
| :<math>{d \over dx} p_n(x) = p_{n-1}(x)</math>
| |
| | |
| instead.
| |
| | |
| ==See also==
| |
| | |
| * [[Sheffer sequence]]
| |
| * [[Umbral calculus]]
| |
| * [[Generalized Appell polynomials]]
| |
| * [[Wick product]]
| |
| | |
| ==References==
| |
| | |
| * {{cite journal|first1=Paul |last1=Appell
| |
| |title=Sur une classe de polynômes
| |
| |journal=Annales scientifiques de l'[[École Normale Supérieure]] 2me série
| |
| |year=1880|volume=9 | pages=119–144|url=http://www.numdam.org/item?id=ASENS_1880_2_9__119_0}}
| |
| * {{cite journal|first1= Steven
| |
| |last1= Roman |first2=Gian-Carlo |last2= Rota
| |
| |title=The Umbral Calculus
| |
| |journal=Advances in Mathematics
| |
| |volume =27|number=2
| |
| |pages=95–188 |year=1978 |doi = 10.1016/0001-8708(78)90087-7 }}.
| |
| * {{cite journal | first1=Gian-Carlo |last1=Rota| first2=D.
| |
| |last2= Kahaner | first3=Andrew |last3= Odlyzko|
| |
| |title=Finite Operator Calculus |pages=685–760
| |
| |journal=Journal of Mathematical Analysis and its Applications
| |
| |volume=42 |number=3|year=1973 | doi=10.1016/0022-247X(73)90172-8}} Reprinted in the book with the same title, Academic Press, New York, 1975.
| |
| * {{cite book | author=Steven Roman | title= The Umbral Calculus | publisher= [[Dover Publications]]}}
| |
| * {{cite book | author=Theodore Seio Chihara | title= An Introduction to Orthogonal Polynomials | publisher= Gordon and Breach, New York | year=1978 | isbn = 0-677-04150-0}}
| |
| | |
| ==External links==
| |
| * {{springer|title=Appell polynomials|id=p/a012800}}
| |
| * [http://mathworld.wolfram.com/AppellSequence.html Appell Sequence] at [[MathWorld]]
| |
| | |
| {{DEFAULTSORT:Appell Sequence}}
| |
| [[Category:Polynomials]]
| |