Fibonacci Polynomial

Author

Eric W. Weisstein
December 21, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/FibonacciPolynomial.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FibonacciPolynomial.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Plot

Plot[Evaluate[Table[Fibonacci[n,x],{n,5}]],{x,-2,2},PlotStyle->{Red,Yellow,Green,Blue,Purple},
AxesLabel->TraditionalForm/@{x,Fibonacci[n,x]}
]

Enumeration

Table[Fibonacci[n,x],{n,0,5}]//ColumnForm//TraditionalForm

CoefficientList[#,x]&/@Table[Fibonacci[n,x],{n,0,6}]

Flatten[%]

Recurrence

RSolve[{F[n+1,x]==x F[n,x]+F[n-1,x],F[1,x]==1,F[2,x]==x},F[n,x],n]

FullSimplify[%,n∈Integers]

Table[(((2^(-n)) (-(((x-(Sqrt[4+(x^2)])))^n)+(((x+(Sqrt[4+(x^2)])))^n)))/(Sqrt[4+(x^2)]))-Fibonacci[n,x],{n,-10,10}]//FullSimplify

Series

(∑_(j=0)^(⌊(n-1)/2⌋))(Binomial[-j+n-1, j]) (x^(-2 j+n-1))==(Subscript[F, n])(x)

Sum[Binomial[n-j-1,j](x^(n-2j-1)),{j,0,Floor[(n-1)/2]}]

FullSimplify[%,n∈Integers]

Table[(((2^(-n)) (-(((x-(Sqrt[4+(x^2)])))^n)+(((x+(Sqrt[4+(x^2)])))^n)))/(Sqrt[4+(x^2)]))-Fibonacci[n,x],{n,-10,10}]//FullSimplify

Generating Function

GeneratingFunction[Fibonacci[n,x],n,t]

Series[-(t/(-1+(t^2)+t x)),{t,0,10}]

Sum[Fibonacci[n,x](t^n),{n,0,10}]

SeriesCoefficient[-(t/(-1+(t^2)+t x)),{t,0,n}]//InputForm

Sums

(∑_(n=0)^∞)(t^n) (Fibonacci2[n, x])==-(t/((t^2)+t x-1))

V6

Sum[Fibonacci[n,x](t^n),{n,0,∞}]//InputForm

V8

Sum[Fibonacci[n,x](t^n),{n,0,∞}]

Identities

(Fibonacci2[n, 1])==(Fibonacci[n])

Fibonacci[n,1]

(Subscript[F, n])((Subscript[U, p-1])(((Sqrt[5])/2)))==((Subscript[F, n p])/(Subscript[F, p])), p odd

FullSimplify[Fibonacci[n,ChebyshevU[p-1,Sqrt[5]/2]],p∈Integers&&p>0&&Mod[p,2]==1&]

Subtract@@@Table[{Fibonacci[n,ChebyshevU[p-1,Sqrt[5]/2]],Fibonacci[n p]/Fibonacci[p]},{p,1,10,2}]//FullSimplify

FullSimplify[%,n∈Integers&&n>0]

Table[%/.n->RandomInteger[{0,10}],{20}]//FullSimplify

Table[ChebyshevU[p-1,Sqrt[5]/2],{p,1,21,2}]

A002878

l=Table[-(Fibonacci[n p]/Fibonacci[p])+Fibonacci[n,ChebyshevU[p-1,Sqrt[5]/2]],{p,1,21,2}]//ColumnForm

l/.n->123

Integral and Derivative

Integral

Integrate[Fibonacci[n,x],x]

Derivative

D[Fibonacci[n,x],x]

Integrate[D[Fibonacci[n,x],x],x]//InputForm

TraditionalForm[%]

Integrate[D[Fibonacci[n,x],x],x]