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| In [[combinatorics]], the '''binomial transform''' is a [[sequence transformation]] (i.e., a transform of a [[sequence]]) that computes its [[forward difference]]s. It is closely related to the '''Euler transform''', which is the result of applying the binomial transform to the sequence associated with its [[ordinary generating function]].
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| ==Definition==
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| The '''binomial transform''', ''T'', of a sequence, {''a''<sub>''n''</sub>}, is the sequence {''s''<sub>''n''</sub>} defined by | |
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| :<math>s_n = \sum_{k=0}^n (-1)^k {n\choose k} a_k.</math>
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| Formally, one may write (''Ta'')<sub>''n''</sub> = ''s''<sub>''n''</sub> for the transformation, where ''T'' is an infinite-dimensional [[operator (mathematics)|operator]] with matrix elements ''T''<sub>''nk''</sub>:
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| :<math>s_n = (Ta)_n = \sum_{k=0}^\infty T_{nk} a_k.</math>
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| The transform is an [[involution (mathematics)|involution]], that is,
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| :<math>TT = 1 \,</math>
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| or, using index notation,
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| :<math>\sum_{k=0}^\infty T_{nk}T_{km} = \delta_{nm}</math>
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| where δ is the [[Kronecker delta function]]. The original series can be regained by
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| :<math>a_n=\sum_{k=0}^n (-1)^k {n\choose k} s_k.</math>
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| The binomial transform of a sequence is just the ''n''th [[forward difference#n-th difference | forward differences]] of the sequence, with odd differences carrying a negative sign, namely:
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| :<math>s_0 = a_0</math>
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| :<math>s_1 = - (\triangle a)_0 = -a_1+a_0</math>
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| :<math>s_2 = (\triangle^2 a)_0 = -(-a_2+a_1)+(-a_1+a_0) = a_2-2a_1+a_0</math>
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| :<math>\vdots\,</math>
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| :<math>s_n = (-1)^n (\triangle^n a)_0</math>
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| where Δ is the [[forward difference operator]].
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| Some authors define the binomial transform with an extra sign, so that it is not self-inverse:
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| :<math>t_n=\sum_{k=0}^n (-1)^{n-k} {n\choose k} a_k</math>
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| whose inverse is
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| :<math>a_n=\sum_{k=0}^n {n\choose k} t_k.</math>
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| ==Example==
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| Binomial transforms can be seen in difference tables. Consider the following:
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| {| style=text-align:center
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| | style="width:9%;"| 0 || style="width:9%;"| || style="width:9%;"| 1 || style="width:9%;"| || style="width:9%;"| 10 || style="width:9%;"| || style="width:9%;"| 63 || style="width:9%;"| || style="width:9%;"| 324 || style="width:9%;"| || style="width:9%;"| 1485
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| |-
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| | || 1 || || 9 || || 53 || || 261 || || 1161
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| |-
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| | || || 8 || || 44 || || 208 || || 900
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| |-
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| | || || || 36 || || 164 || || 692
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| |-
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| | || || || || 128 || || 528
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| |-
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| | || || || || || 400
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| |}
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| The top line 0, 1, 10, 63, 324, 1485,... (a sequence defined by (2''n''<sup>2</sup> + ''n'')3<sup>''n'' − 2</sup>) is the (noninvolutive version of the) binomial transform of the diagonal 0, 1, 8, 36, 128, 400,... (a sequence defined by ''n''<sup>2</sup>2<sup>''n'' − 1</sup>).
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| ==Shift states==
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| The binomial transform is the [[shift operator]] for the [[Bell number]]s. That is,
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| :<math>B_{n+1}=\sum_{k=0}^n {n\choose k} B_k</math>
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| where the ''B''<sub>''n''</sub> are the Bell numbers.
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| ==Ordinary generating function==
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| The transform connects the [[generating function]]s associated with the series. For the [[ordinary generating function]], let
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| :<math>f(x)=\sum_{n=0}^\infty a_n x^n</math>
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| and
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| :<math>g(x)=\sum_{n=0}^\infty s_n x^n </math>
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| then
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| :<math>g(x) = (Tf)(x) = \frac{1}{1-x} f\left(\frac{x}{x-1}\right).</math>
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| ==Euler transform==
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| The relationship between the ordinary generating functions is sometimes called the '''Euler transform'''. It commonly makes its appearance in one of two different ways. In one form, it is used to [[series acceleration|accelerate the convergence]] of an [[alternating series]]. That is, one has the identity
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| :<math>\sum_{n=0}^\infty (-1)^n a_n = \sum_{n=0}^\infty (-1)^n
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| \frac {\Delta^n a_0} {2^{n+1}}</math>
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| which is obtained by substituting ''x''=1/2 into the last formula above. The terms on the right hand side typically become much smaller, much more rapidly, thus allowing rapid numerical summation.
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| The Euler transform can be generalized (Borisov B. and Shkodrov V., 2007):
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| :<math>\sum_{n=0}^\infty (-1)^n {n+p\choose n} a_n = \sum_{n=0}^\infty (-1)^n
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| {n+p\choose n}\frac {\Delta^n a_0} {2^{n+p+1}}</math>,
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| where ''p'' = 0, 1, 2,...
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| The Euler transform is also frequently applied to the [[Euler hypergeometric integral]] <math>\,_2F_1</math>. Here, the Euler transform takes the form:
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| :<math>\,_2F_1 (a,b;c;z) = (1-z)^{-b} \,_2F_1 \left(c-a, b; c;\frac{z}{z-1}\right).</math>
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| The binomial transform, and its variation as the Euler transform, is notable for its connection to the [[continued fraction]] representation of a number. Let <math>0 < x < 1</math> have the continued fraction representation
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| :<math>x=[0;a_1, a_2, a_3,\cdots]</math>
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| then
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| :<math>\frac{x}{1-x}=[0;a_1-1, a_2, a_3,\cdots]</math>
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| and
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| :<math>\frac{x}{1+x}=[0;a_1+1, a_2, a_3,\cdots].</math>
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| ==Exponential generating function==
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| For the [[exponential generating function]], let
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| :<math>\overline{f}(x)= \sum_{n=0}^\infty a_n \frac{x^n}{n!}</math>
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| and
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| :<math>\overline{g}(x)= \sum_{n=0}^\infty s_n \frac{x^n}{n!}</math>
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| then
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| :<math>\overline{g}(x) = (T\overline{f})(x) = e^x \overline{f}(-x).</math>
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| The [[Borel summation|Borel transform]] will convert the ordinary generating function to the exponential generating function.
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| ==Integral representation==
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| When the sequence can be interpolated by a [[complex analytic]] function, then the binomial transform of the sequence can be represented by means of a [[Nörlund–Rice integral]] on the interpolating function.
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| ==Generalizations==
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| Prodinger gives a related, [[modular form|modular-like]] transformation: letting
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| :<math>u_n = \sum_{k=0}^n {n\choose k} a^k (-c)^{n-k} b_k</math>
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| gives
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| :<math>U(x) = \frac{1}{cx+1} B\left(\frac{ax}{cx+1}\right)</math>
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| where ''U'' and ''B'' are the ordinary generating functions associated with the series <math>\{u_n\}</math> and <math>\{b_n\}</math>, respectively.
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| The rising ''k''-binomial transform is sometimes defined as
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| :<math>\sum_{j=0}^n {n\choose j} j^k a_j.</math>
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| The falling ''k''-binomial transform is
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| :<math>\sum_{j=0}^n {n\choose j} j^{n-k} a_j</math>.
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| Both are homomorphisms of the [[kernel (algebra)|kernel]] of the [[Hankel transform of a series]].
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| In the case where the binomial transform is defined as
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| :<math>\sum_{i=0}^n(-1)^{n-i}\binom{n}{i}a_i=b_n.</math>
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| Let this be equal to the function <math>\mathfrak J(a)_n=b_n.</math>
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| If a new [[forward difference]] table is made and the first elements from each row of this table are taken to form a new sequence <math>\{b_n\}</math>, then the second binomial transform of the original sequence is,
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| :<math>\mathfrak J^2(a)_n=\sum_{i=0}^n(-2)^{n-i}\binom{n}{i}a_i.</math>
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| If the same process is repeated ''k'' times, then it follows that,
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| :<math>\mathfrak J^k(a)_n=b_n=\sum_{i=0}^n(-k)^{n-i}\binom{n}{i}a_i.</math>
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| Its inverse is,
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| :<math>\mathfrak J^{-k}(b)_n=a_n=\sum_{i=0}^nk^{n-i}\binom{n}{i}b_i.</math>
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| This can be generalized as,
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| :<math>\mathfrak J^k(a)_n=b_n=(\mathbf E-k)^na_0</math>
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| where <math>\mathbf E</math> is the [[shift operator]].
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| Its inverse is
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| :<math>\mathfrak J^{-k}(b)_n=a_n=(\mathbf E+k)^nb_0.</math>
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| ==See also==
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| * [[Newton series]]
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| * [[Hankel matrix]]
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| * [[Möbius transform]]
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| * [[Stirling transform]]
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| * [[Euler summation]]
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| * [[List of factorial and binomial topics]]
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| ==References==
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| * John H. Conway and Richard K. Guy, 1996, ''The Book of Numbers''
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| * Donald E. Knuth, ''The Art of Computer Programming Vol. 3'', (1973) Addison-Wesley, Reading, MA.
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| * Helmut Prodinger, 1992, ''[http://math.sun.ac.za/~prodinger/abstract/abs_87.htm Some information about the Binomial transform]''
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| * Michael Z. Spivey and Laura L. Steil, 2006, ''[http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.pdf The k-Binomial Transforms and the Hankel Transform]''
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| * Borisov B. and Shkodrov V., 2007, Divergent Series in the Generalized Binomial Transform, Adv. Stud. Cont. Math., 14 (1): 77-82
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| ==External links==
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| *[http://mathworld.wolfram.com/BinomialTransform.html Binomial Transform],
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| [[Category:Transforms]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Hypergeometric functions]]
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