en>BD2412: /* Definition */Fixing links to disambiguation pages using AWB

2013-12-27T22:52:51Z

Definition: Fixing links to disambiguation pages using AWB

← Older revision		Revision as of 00:52, 28 December 2013
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	The ~~name~~ of the ~~writer~~ is ~~Luther~~. ~~Meter studying~~ is ~~exactly where my primary income arrives from but quickly I~~'~~ll be on my own~~. The ~~factor she adores most~~ is ~~to perform handball but she~~ can'~~t make it her occupation~~. ~~Alabama has always been his house~~ and ~~his family enjoys it.~~<br><br>~~Also visit my blog post~~ :: ~~extended car warranty~~ ([http://~~Lahnsinfonie~~.de/~~index~~.~~php?mod=users&action=view&id=18009 his response~~])		The following examples are in the spirit of [[George Pólya]], who advocated learning mathematics by doing and re-capitulating as many examples and proofs as possible. The purpose of this article is to present common tricks of the trade in context, so that people may incorporate them into their knowledge.

			== Worked example A: basics ==
			New generating functions can be created by extending simpler generating functions. For [[Worked-example effect\|example]], starting with

			:<math>G(1;x)=\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}</math>

			and replacing <math>x</math> with <math>ax</math>, we obtain

			:<math>G(1;ax)=\frac{1}{1-ax} = 1+(ax)+(ax)^2+\cdots+(ax)^n+\cdots =\sum_{n=0}^{\infty} a^n x^n = G(a^n;x).</math>

			=== Bivariate generating functions ===
			One can define generating functions in several variables, for series with several indices. These are often called '''super generating functions,''' and for 2 variables are often called '''bivariate generating functions.'''

			For instance, since <math>(1+x)^n</math> is the generating function for [[binomial coefficients]] for a fixed ''n,'' one may ask for a bivariate generating function that generates the binomial coefficients <math>\binom{n}{k}</math> for all ''k'' and ''n.''
			To do this, consider <math>(1+x)^n</math> as ''itself'' a series (in ''n''), and find the generating function in ''y'' that has these as coefficients. Since the generating function for <math>a^n</math> is just <math>1/(1-ay)</math>, the generating function for the binomial coefficients is:
			:<math>\frac{1}{1-(1+x)y}=1+(1+x)y+(1+x)^2y^2+\dots,</math>
			and the coefficient on <math>x^ky^n</math> is the <math>\binom{n}{k}</math> binomial coefficient.

			== Worked example B: Fibonacci numbers ==

			Consider the problem of finding a [[Closed-form expression\|closed formula]] for the [[Fibonacci number]]s ''F''<sub>''n''</sub> defined by ''F''<sub>0</sub> = 0, ''F''<sub>1</sub> = 1, and ''F''<sub>''n''</sub> = ''F''<sub>''n''−1</sub> + ''F''<sub>''n''−2</sub> for ''n'' ≥ 2. We form the ordinary generating function

			:<math>
			f = \sum_{n \ge 0} F_n x^n
			</math>

			for this sequence. The generating function for the sequence (''F''<sub>''n''−1</sub>) is ''xf'' and that of (''F''<sub>''n''−2</sub>) is ''x''<sup>2</sup>''f''. From the recurrence relation, we therefore see that the power series ''xf'' + ''x''<sup>2</sup>''f'' agrees with ''f'' except for the first two coefficients:

			:<math>
			\begin{array}{rcrcrcrcrcrcr}
			f & = & F_0x^0 & + & F_1x^1 & + & F_2x^2 & + & \cdots & + & F_ix^i & + &\cdots\\
			xf & = & & & F_0x^1 & + & F_1x^2 & + & \cdots & + &F_{i-1}x^i & + &\cdots\\
			x^2f & = & & & & & F_0x^2 & + & \cdots & + &F_{i-2}x^i & +&\cdots\\
			(x+x^2)f & = & & & F_0x^1 & + & (F_0+F_1)x^2 & + & \cdots & + & (F_{i-1}+F_{i-2})x^i & +&\cdots\\
			& = & & & & & F_2x^2 & + & \cdots & + & F_ix^i & +& \cdots\\
			\end{array}
			</math>

			Taking these into account, we find that

			:<math>
			f = xf + x^2 f + x . \,\!
			</math>

			(This is the crucial step; recurrence relations can almost always be translated into equations for the generating functions.) Solving this equation for ''f'', we get

			:<math>
			f = \frac{x} {1 - x - x^2} .
			</math>

			The denominator can be factored using the [[golden ratio]] φ<sub>1</sub> = (1 + √5)/2 and φ<sub>2</sub> = (1 − √5)/2, and the technique of [[partial fraction decomposition]] yields

			:<math>
			f = \frac{1}{\sqrt{5}} \left (\frac{1}{1-\varphi_1 x} - \frac{1} {1- \varphi_2 x} \right ) .
			</math>

			These two formal power series are known explicitly because they are [[geometric series]]; comparing coefficients, we find the explicit formula

			:<math>
			F_n = \frac{1} {\sqrt{5}} (\varphi_1^n - \varphi_2^n).
			</math>

			==External links==
			* [http://www.cut-the-knot.org/ctk/GeneratingFunctions.shtml Generating Functions, Power Indices and Coin Change] at [[cut-the-knot]]
			* [http://www.math.upenn.edu/~wilf/gfologyLinked2.pdf Generatingfunctionology (PDF)]

			{{DEFAULTSORT:Examples Of Generating Functions}}
			[[Category:Generating functions]]
			[[Category:Mathematical examples\|Generating functions]]

en>FrescoBot: Bot: links syntax

2010-10-24T06:55:04Z

Bot: links syntax

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The name of the writer is Luther. Meter studying is exactly where my primary income arrives from but quickly I'll be on my own. The factor she adores most is to perform handball but she can't make it her occupation. Alabama has always been his house and his family enjoys it.<br><br>Also visit my blog post :: extended car warranty ([http://Lahnsinfonie.de/index.php?mod=users&action=view&id=18009 his response])

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en>BD2412: /* Definition */Fixing links to disambiguation pages using AWB

en>FrescoBot: Bot: links syntax