# Find a generating function with Fibonacci

$$G(x) = \sum_{n=1}^\infty na_n x^n$$

Hello. I need to find a generating function for the summation above, where $a_n$ is the Fibonacci sequence.

I have found the generating function for the fib itself but am confused as to how I can progress further. I also tried plugging in the closed form of the fib sequence and tried to separate out variables to get a sum of sigmas, but it didn't help. Can someone guide me in the right direction?

• hint: differentiate the generating function for the fibonacci numbers, then correct for the power of $x$. – lulu Feb 26 '16 at 16:46
• Hint: $f(z) = \sum_{k=0}^\infty a_k z^k\quad\implies\quad z\frac{df(z)}{dz} = \sum_{k=1}^\infty ka_k z^k$. – achille hui Feb 26 '16 at 16:48
• A very good solution is given here. – Hypergeometricx Feb 26 '16 at 17:04
• Please transcribe the image into proper MathJax. Many here (myself included) won't click on random links. – vonbrand Feb 26 '16 at 18:03

Note that $$G(x)=\sum_{n=1}^{\infty}na_nx^n=x\sum_{n=1}^{\infty}na_nx^{n-1}=x\frac{d}{dx}\left[\sum_{n=0}^{\infty}a_nx^n\right]$$
So, if you've already got the generating function for the Fibonacci sequence $(a_n)_{n=0}^{\infty}$, you can easily modify it to get the function that you're looking for.