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

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?

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

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.