How can I write an algorithm in maple that gives me a lisft of the Fibonacci polynomies between two numbers using the recursion:

F1(x) = 1

F2(x) = x

Fn+1(x) = x*Fn(x) + Fn−1(x)

if anyone knows please help me!!:)

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- Dec 21st 2007, 01:03 AMmaplemaple[SOLVED] Maple - HELP PLEASE!
How can I write an algorithm in maple that gives me a lisft of the Fibonacci polynomies between two numbers using the recursion:

F1(x) = 1

F2(x) = x

Fn+1(x) = x*Fn(x) + Fn−1(x)

if anyone knows please help me!!:) - Dec 22nd 2007, 02:03 PMgalactus
Yes, it's the rsolve command.

i.e., to generate the closed-form for the nth Fibonacci number type:

**rsolve({f(n)=f(n-1)+f(n-2),f(0)=1,f(1)=1},{f(n)});**

This will give you:

$\displaystyle f(n)=(\frac{1}{2}+\frac{1}{10}\sqrt{5})(\frac{1}{2 }\sqrt{5}+\frac{1}{2})^{n}+(\frac{1}{2}-\frac{1}{10}\sqrt{5})(\frac{-1}{2}\sqrt{5}+\frac{1}{2})^{n}$

Which is the formula we get from Binet's formula.

By tweaking it accordingly, you should get the polynomials:

$\displaystyle x+x^{2}+2x^{3}+3x^{4}+5x^{5}+..........$

The closed-form is $\displaystyle \boxed{\frac{x}{1-x-x^{2}}}$