147.177.42.109: /* Closure operators on partially ordered sets */ Fixed typo in ref

2013-09-10T17:15:29Z

Closure operators on partially ordered sets: Fixed typo in ref

← Older revision		Revision as of 19:15, 10 September 2013
Line 1:		Line 1:
	~~Hello~~, ~~dear friend! I am Weldon~~. ~~I smile that I could unify~~ to the ~~entire globe~~. ~~I live~~ in ~~Germany~~, in the ~~NI region~~. ~~I dream to visit~~ the ~~different countries~~, ~~to obtain acquainted~~ with ~~fascinating people~~.<br><br>~~Take~~ a ~~look at my~~ page :~~: FIFA 15 coin hack~~ [[~~http://www~~.~~fifa15~~-~~coingenerator~~.~~com/ find out here now~~]]		In [[mathematics]], the '''Fibonacci polynomials''' are a [[polynomial sequence]] which can be considered as a generalization of the [[Fibonacci number]]s. The polynomials generated in a similar way from the [[Lucas numbers]] are called '''Lucas polynomials'''.

			==Definition==
			These Fibonacci [[polynomial]]s are defined by a [[recurrence relation]]:<ref name=BQ141>Benjamin & Quinn p. 141</ref>

			:<math>F_n(x)= \begin{cases}
			0, & \mbox{if } n = 0\\
			1, & \mbox{if } n = 1\\
			x F_{n - 1}(x) + F_{n - 2}(x),& \mbox{if } n \geq 2
			\end{cases}</math>

			The first few Fibonacci polynomials are:
			:<math>F_0(x)=0 \,</math>
			:<math>F_1(x)=1 \,</math>
			:<math>F_2(x)=x \,</math>
			:<math>F_3(x)=x^2+1 \,</math>
			:<math>F_4(x)=x^3+2x \,</math>
			:<math>F_5(x)=x^4+3x^2+1 \,</math>
			:<math>F_6(x)=x^5+4x^3+3x \,</math>

			The Lucas polynomials use the same recurrence with different starting values:<ref>Benjamin & Quinn p. 142</ref>
			<math>L_n(x) = \begin{cases}
			2, & \mbox{if } n = 0 \\
			x, & \mbox{if } n = 1 \\
			x L_{n - 1}(x) + L_{n - 2}(x), & \mbox{if } n \geq 2.
			\end{cases}</math>

			The first few Lucas polynomials are:

			:<math>L_0(x)=2 \,</math>
			:<math>L_1(x)=x \,</math>
			:<math>L_2(x)=x^2+2 \,</math>
			:<math>L_3(x)=x^3+3x \,</math>
			:<math>L_4(x)=x^4+4x^2+2 \,</math>
			:<math>L_5(x)=x^5+5x^3+5x \,</math>
			:<math>L_6(x)=x^6+6x^4+9x^2 + 2. \,</math>

			The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at ''x'' = 1; [[Pell numbers]] are recovered by evaluating ''F''<sub>''n''</sub> at ''x'' = 2. The degrees of ''F''<sub>''n''</sub> is ''n'' − 1 and the degree of ''L''<sub>''n''</sub> is ''n''. The [[Generating function#Ordinary generating function\|ordinary generating function]] for the sequences are:<ref>{{MathWorld \| urlname=FibonacciPolynomial \| title=Fibonacci Polynomial}}</ref>

			:<math> \sum_{n=0}^\infty F_n(x) t^n = \frac{t}{1-xt-t^2}</math>
			:<math> \sum_{n=0}^\infty L_n(x) t^n = \frac{2-xt}{1-xt-t^2}.</math>

			The polynomials can be expressed in terms of [[Lucas sequence]]s as
			:<math>F_n(x) = U_n(x,-1),\,</math>
			:<math>L_n(x) = V_n(x,-1).\,</math>

			==Identities==
			{{main\|Lucas sequence}}
			As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities.

			First, they can be defined for negative indices by<ref name=Springer>Springer</ref>
			:<math>F_{-n}(x)=(-1)^{n-1}F_{n}(x),\,L_{-n}(x)=(-1)^nL_{n}(x).</math>
			Other identities include:<ref name=Springer/>
			:<math>F_{m+n}(x)=F_{m+1}(x)F_n(x)+F_m(x)F_{n-1}(x)\,</math>
			:<math>L_{m+n}(x)=L_m(x)L_n(x)-(-1)^nL_{m-n}(x)\,</math>
			:<math>F_{n+1}(x)F_{n-1}(x)- F_n(x)^2=(-1)^n\,</math>
			:<math>F_{2n}(x)=F_n(x)L_n(x).\,</math>
			Closed form expressions, similar to Binet's formula are:<ref name=Springer/>
			:<math>F_n(x)=\frac{\alpha(x)^n-\beta(x)^n}{\alpha(x)-\beta(x)},\,L_n(x)=\alpha(x)^n+\beta(x)^n,</math>
			where
			:<math>\alpha(x)=\frac{x+\sqrt{x^2+4}}{2},\,\beta(x)=\frac{x-\sqrt{x^2+4}}{2}</math>
			are the solutions (in ''t'') of
			:<math>t^2-xt-1=0.\,</math>

			==Combinatorial interpretation==
			[[File:PascalTriangleFibanacci.svg\|thumb\|right\|360px\|The coefficients of the Fibonacci polynomials can be read off from Pascal's triangle following the "shallow" diagonals (shown in red). The sums of the coefficients are the Fibonacci numbers.]]
			If ''F''(''n'',''k'') is the coefficient of ''x<sup>k</sup>'' in ''F<sub>n</sub>''(''x''), so
			:<math>F_n(x)=\sum_{k=0}^n F(n,k)x^k,\,</math>
			then ''F''(''n'',''k'') is the number of ways an ''n''−1 by 1 rectangle can be tiled with 2 by 1 [[domino]]es and 1 by 1 squares so that exactly ''k'' squares are used.<ref name=BQ141/> Equivalently, ''F''(''n'',''k'') is the number of ways of writing ''n''−1 as an [[Composition (number theory)\|ordered sum]] involving only 1 and 2, so that 1 is used exactly ''k'' times. For example F(6,3)=4 and 5 can be written in 4 ways, 1+1+1+2, 1+1+2+1, 1+2+1+1, 2+1+1+1, as a sum involving only 1 and 2 with 1 used 3 times. By counting the number of times 1 and 2 are both used in such a sum, it is evident that ''F''(''n'',''k'') is equal to the [[binomial coefficient]]
			:<math>F(n,k)=\binom{\tfrac{n+k-1}{2}}{k}</math>
			when ''n'' and ''k'' have opposite parity. This gives a way of reading the coefficients from [[Pascal's triangle]] as shown on the right.

			==References==
			{{reflist}}
			*{{cite book \|title=Proofs that Really Count
			\|first1=Arthur T.\|last1=Benjamin\|first2=Jennifer J.\|last2=Quinn
			\|publisher=MAA\|year=2003\|isbn=0-88385-333-7
			\|chapter=§9.4 Fibonacci and Lucas Polynomial\|page=141}}
			<!-- Note: This source idiosyncratically shifts the index by 1, allow for this when checking the formulas. -->
			*{{SpringerEOM\|title=Fibonacci polynomials
			\|id=Fibonacci_polynomials&oldid=14185\|last=Philippou\|first=Andreas N.}}
			*{{SpringerEOM\|title=Lucas polynomials
			\|id=Lucas_polynomials&oldid=17297\|last=Philippou\|first=Andreas N.}}
			* {{MathWorld \| urlname=LucasPolynomial\| title=Lucas Polynomial}}

			==Further reading==
			* {{cite journal \| first1=V. E.\|last1=Hoggatt \| authorlink=Verner Emil Hoggatt, Jr. \| first2=Marjorie \| last2=Bicknell \| title=Roots of Fibonacci polynomials. \| journal=[[Fibonacci Quarterly]] \| volume=11 \| pages=271–274 \| year=1973 \| issn=0015-0517\| mr=0332645 }}
			* {{cite journal \| first1=V. E.\|last1=Hoggatt \| first2=Calvin T. \| last2=Long \| title=Divisibility properties of generalized Fibonacci Polynomials \| journal=[[Fibonacci Quarterly]] \| volume=12 \| page=113 \| year=1974 \| mr=0352034 }}
			* {{cite journal \| last1=Ricci \|first1=Paolo Emilio \| title=Generalized Lucas polynomials and Fibonacci polynomials \| journal=Rivista di Matematica della Università di Parma\|series= V. Ser. \| volume=4 \| year=1995 \| pages=137–146 \|mr=1395332 }}
			* {{cite journal\|last1=Yuan \|first1=Yi \|last2=Zhang\|first2=Wenpeng \|journal=Fibonacci Quarterly\| year=2002 \|title =Some identities involving the Fibonacci Polynomials \|page=314\|mr=1920571\|volume=40\|issue=4}}
			* {{cite journal\|first1=Johann\|last1=Cigler \|journal=Fibonacci Quarterly \|year=2003 \|mr=1962279 \| pages=31–40\|number=41\|title=q-Fibonacci polynomials}}

			==External links==
			*{{SloanesRef \|sequencenumber=A162515\|name=Triangle of coefficients of polynomials defined by Binet form...}}
			*{{SloanesRef \|sequencenumber=A011973\|name=Triangle of coefficients of Fibonacci polynomials.}}

			[[Category:Polynomials]]
			[[Category:Fibonacci numbers]]

98.122.165.172 at 20:54, 14 August 2012

2012-08-14T20:54:19Z

New page

Hello, dear friend! I am Weldon. I smile that I could unify to the entire globe. I live in Germany, in the NI region. I dream to visit the different countries, to obtain acquainted with fascinating people.<br><br>Take a look at my page :: FIFA 15 coin hack [[http://www.fifa15-coingenerator.com/ find out here now]]

Closure operator - Revision history

147.177.42.109: /* Closure operators on partially ordered sets */ Fixed typo in ref

98.122.165.172 at 20:54, 14 August 2012