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| | I am 23 years old and my name is Irene Toro. I life in Bodo (Norway).<br><br>Look at my web site; [http://hemorrhoidtreatmentfix.com/prolapsed-hemorrhoid-treatment prolapsed hemorrhoids treatment] |
| In [[mathematics]], a [[polynomial sequence]], i.e., a sequence of [[polynomial]]s indexed by { 0, 1, 2, 3, ... } in which the index of each polynomial equals its degree, is said to be of '''binomial type''' if it satisfies the sequence of identities
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| :<math>p_n(x+y)=\sum_{k=0}^n{n \choose k}\, p_k(x)\, p_{n-k}(y).</math>
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| Many such sequences exist. The set of all such sequences forms a [[Lie group]] under the operation of umbral composition, explained below. Every sequence of binomial type may be expressed in terms of the [[Bell polynomial]]s. Every sequence of binomial type is a [[Sheffer sequence]] (but most Sheffer sequences are not of binomial type). Polynomial sequences put on firm footing the vague 19th century notions of [[umbral calculus]].
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| ==Examples==
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| * In consequence of this definition the [[binomial theorem]] can be stated by saying that the sequence { ''x''<sup>''n''</sup> : ''n'' = 0, 1, 2, ... } is of binomial type.
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| * The sequence of "[[lower factorial]]s" is defined by
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| ::<math>(x)_n=x(x-1)(x-2)\cdot\cdots\cdot(x-n+1).</math>
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| :(In the theory of special functions, this same notation denotes [[upper factorial]]s, but this present usage is universal among [[combinatorics|combinatorialists]].) The product is understood to be 1 if ''n'' = 0, since it is in that case an [[empty product]]. This polynomial sequence is of binomial type.
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| * Similarly the "[[upper factorial]]s"
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| ::<math>x^{(n)}=x(x+1)(x+2)\cdot\cdots\cdot(x+n-1)</math>
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| :are a polynomial sequence of binomial type.
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| * The [[Abel polynomials]]
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| ::<math>p_n(x)=x(x-an)^{n-1} \,</math>
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| :are a polynomial sequence of binomial type.
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| * The [[Touchard polynomials]]
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| ::<math>p_n(x)=\sum_{k=1}^n S(n,k)x^k</math>
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| :where ''S''(''n'', ''k'') is the number of partitions of a set of size ''n'' into ''k'' disjoint non-empty subsets, is a polynomial sequence of binomial type. [[Eric Temple Bell]] called these the "exponential polynomials" and that term is also sometimes seen in the literature. The coefficients ''S''(''n'', ''k'' ) are "[[Stirling number]]s of the second kind". This sequence has a curious connection with the [[Poisson distribution]]: If ''X'' is a [[random variable]] with a Poisson distribution with expected value λ then E(''X''<sup>''n''</sup>) = ''p''<sub>''n''</sub>(λ). In particular, when λ = 1, we see that the ''n''th moment of the Poisson distribution with expected value 1 is the number of partitions of a set of size ''n'', called the ''n''th [[Bell numbers|Bell number]]. This fact about the ''n''th moment of that particular Poisson distribution is "[[Bell numbers|Dobinski's formula]]".
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| ==Characterization by delta operators==
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| It can be shown that a polynomial sequence { ''p''<sub>''n''</sub>(x) : ''n'' = 0, 1, 2, ... } is of binomial type if and only if all three of the following conditions hold:
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| * The [[linear transformation]] on the space of polynomials in ''x'' that is characterized by
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| ::<math>p_n(x)\mapsto np_{n-1}(x)</math>
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| :is [[shift-equivariant]], and
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| * ''p''<sub>0</sub>(''x'') = 1 for all ''x'', and
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| * ''p''<sub>''n''</sub>(0) = 0 for ''n'' > 0.
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| (The statement that this operator is shift-equivariant is the same as saying that the polynomial sequence is a [[Sheffer sequence]]; the set of sequences of binomial type is properly included within the set of Sheffer sequences.)
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| ===Delta operators===
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| That linear transformation is clearly a [[delta operator]], i.e., a shift-equivariant linear transformation on the space of polynomials in ''x'' that reduces degrees of polynomials by 1. The most obvious examples of delta operators are [[difference operator]]s and differentiation. It can be shown that every delta operator can be written as a [[power series]] of the form
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| :<math>Q=\sum_{n=1}^\infty c_n D^n</math>
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| where ''D'' is differentiation (note that the lower bound of summation is 1). Each delta operator ''Q'' has a unique sequence of "basic polynomials", i.e., a polynomial sequence satisfying
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| #<math>p_0(x)=1, \,</math>
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| #<math>p_n(0)=0\quad{\rm for\ }n\geq 1,{\rm\ and}</math>
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| #<math>Qp_n(x)=np_{n-1}(x). \,</math>
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| It was shown in 1973 by [[Gian-Carlo Rota|Rota]], Kahaner, and [[Andrew Odlyzko|Odlyzko]], that a polynomial sequence is of binomial type if and only if it is the sequence of basic polynomials of some delta operator. Therefore, this paragraph amounts to a recipe for generating as many polynomial sequences of binomial type as one may wish.
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| ==Characterization by Bell polynomials==
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| For any sequence ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... of scalars, let
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| :<math>p_n(x)=\sum_{k=1}^n B_{n,k}(a_1,\dots,a_{n-k+1}) x^k.</math>
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| Where ''B''<sub>''n'',''k''</sub>(''a''<sub>1</sub>, ..., ''a''<sub>''n''−''k''+1</sub>) is the [[Bell polynomials|Bell polynomial]]. Then this polynomial sequence is of binomial type. Note that for each ''n'' ≥ 1,
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| :<math>p_n'(0)=a_n.\,</math>
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| Here is the main result of this section:
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| '''Theorem:''' All polynomial sequences of binomial type are of this form.
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| A result in Mullin and Rota, repeated in Rota, Kahaner, and Odlyzko (see ''References'' below) states that every polynomial sequence { ''p''<sub>''n''</sub>(''x'') }<sub>''n''</sub> of binomial type is determined by the sequence { ''p''<sub>''n''</sub>′(0) }<sub>''n''</sub>, but those sources do not mention Bell polynomials.
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| This sequence of scalars is also related to the delta operator. Let
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| :<math>P(t)=\sum_{n=1}^\infty {a_n \over n!} t^n.</math>
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| Then
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| :<math>P^{-1}\left({d \over dx}\right)\,</math>
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| is the delta operator of this sequence.
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| ==Characterization by a convolution identity==
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| For sequences ''a''<sub>''n''</sub>, ''b''<sub>''n''</sub>, ''n'' = 0, 1, 2, ..., define a sort of [[convolution]] by
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| :<math>(a \diamondsuit b)_n = \sum_{j=0}^n {n \choose j} a_j b_{n-j}.</math>
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| Let <math>a_n^{k\diamondsuit}\,</math> be the ''n''th term of the sequence
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| :<math>\underbrace{a\diamondsuit\cdots\diamondsuit a}_{k\text{ factors}}.\,</math>
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| Then for any sequence ''a''<sub>''i''</sub>, ''i'' = 0, 1, 2, ..., with ''a''<sub>0</sub> = 0, the sequence defined by ''p''<sub>0</sub>(''x'') = 1 and
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| :<math>p_n(x) = \sum_{k=1}^n {a_{n}^{k\diamondsuit} x^k \over k!}\,</math>
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| for ''n'' ≥ 1, is of binomial type, and every sequence of binomial type is of this form. This result is due to Alessandro di Bucchianico (see '''References''' below).
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| ==Characterization by generating functions==
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| Polynomial sequences of binomial type are precisely those whose [[generating function]]s are formal (not necessarily convergent) [[power series]] of the form
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| :<math>\sum_{n=0}^\infty {p_n(x) \over n!}t^n=e^{xf(t)}</math>
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| where ''f''(''t'') is a formal power series whose [[constant term]] is zero and whose first-degree term is not zero. It can be shown by the use of the power-series version of [[Faà di Bruno's formula]] that
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| :<math>f(t)=\sum_{n=1}^\infty {p_n\,'(0) \over n!}t^n.</math>
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| The delta operator of the sequence is ''f''<sup>−1</sup>(''D''), so that
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| :<math>f^{-1}(D)p_n(x)=np_{n-1}(x).</math>
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| ===A way to think about these generating functions===
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| The coefficients in the product of two formal power series
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| :<math>\sum_{n=0}^\infty {a_n \over n!}t^n</math>
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| and
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| :<math>\sum_{n=0}^\infty {b_n \over n!}t^n</math>
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| are
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| :<math>c_n=\sum_{k=0}^n {n \choose k} a_k b_{n-k}</math>
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| (see also [[Cauchy product]]). If we think of ''x'' as a parameter indexing a family of such power series, then the binomial identity says in effect that the power series indexed by ''x'' + ''y'' is the product of those indexed by ''x'' and by ''y''. Thus the ''x'' is the argument to a function that maps sums to products: an [[exponential function]]
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| :<math>g(t)^x=e^{x f(t)}</math>
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| where ''f''(''t'') has the form given above.
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| ==Umbral composition of polynomial sequences==
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| The set of all polynomial sequences of binomial type is a [[group (mathematics)|group]] in which the group operation is "umbral composition" of polynomial sequences. That operation is 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, and
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| :<math>p_n(x)=\sum_{k=0}^n a_{n,k}\, x^k.</math>
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| Then the umbral composition ''p'' o ''q'' 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)</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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| With the delta operator defined by a power series in ''D'' as above, the natural bijection between delta operators and polynomial sequences of binomial type, also defined above, is a group isomorphism, in which the group operation on power series is formal composition of formal power series.
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| ==Cumulants and moments==
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| The sequence κ<sub>''n''</sub> of coefficients of the first-degree terms in a polynomial sequence of binomial type may be termed the [[cumulant]]s of the polynomial sequence. It can be shown that the whole polynomial sequence of binomial type is determined by its cumulants, in a way discussed in the article titled ''[[cumulant]]''. Thus
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| :<math> p_n'(0)=\kappa_n= \,</math> the ''n''th cumulant
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| and
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| :<math> p_n(1)=\mu_n'= \,</math> the ''n''th moment.
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| These are "formal" cumulants and "formal" [[moment (mathematics)|moments]], as opposed to cumulants of a [[probability distribution]] and moments of a probability distribution.
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| Let
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| :<math>f(t)=\sum_{n=1}^\infty\frac{\kappa_n}{n!}t^n</math>
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| be the (formal) cumulant-generating function. Then
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| :<math>f^{-1}(D) \,</math>
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| is the delta operator associated with the polynomial sequence, i.e., we have
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| :<math>f^{-1}(D)p_n(x)=np_{n-1}(x). \,</math>
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| ==Applications==
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| The concept of binomial type has applications in [[combinatorics]], [[probability]], [[statistics]], and a variety of other fields.
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| ==See also==
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| * [[List of factorial and binomial topics]]
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| * [[Binomial-QMF]] (Daubechies wavelet filters)
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| ==References==
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| * [[Gian-Carlo Rota|G.-C. Rota]], D. Kahaner, and [[Andrew Odlyzko|A. Odlyzko]], "Finite Operator Calculus," ''Journal of Mathematical Analysis and its Applications'', vol. 42, no. 3, June 1973. Reprinted in the book with the same title, Academic Press, New York, 1975.
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| * R. Mullin and G.-C. Rota, "On the Foundations of Combinatorial Theory III: Theory of Binomial Enumeration," in ''Graph Theory and Its Applications'', edited by Bernard Harris, Academic Press, New York, 1970.
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| As the title suggests, the second of the above is explicitly about applications to [[combinatorics|combinatorial]] enumeration.
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| * di Bucchianico, Alessandro. ''Probabilistic and Analytical Aspects of the Umbral Calculus'', Amsterdam, [[Centrum Wiskunde & Informatica|CWI]], 1997.
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| * {{mathworld|urlname=Binomial-TypeSequence|title=Binomial-Type Sequence}}
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| [[Category:Polynomials]]
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| [[Category:Factorial and binomial topics]]
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