# Is there any mathematic meaning of generating function at some value?

Out of curiosity I am trying to learn some material about generating functions. Now I understand that if I will expand Fibonacci generating function, $f(x) = \frac{1}{1-x-x^2}$ I will get a series where coefficients are Fibonacci numbers.

What I can't understand is whether there is any mathematical meaning $f(a)$ where $a$ is some value? From what I learnt so far (I hope that I understood this correctly) it looks like there is some meaning for any x that is in the convergence radius ($\frac{2}{1+ \sqrt{5}}$).

So what is the mathematical meaning of f(1.5) or f(0) or f(0.5) for example?

• First of all, this only converges for $|x|<\frac{1}{\varphi} = \frac{2}{1+\sqrt{5}}$, not $|x|<\varphi$ as you state. To answer the key part of the question, how would you define the Fibonacci sequence? Do you simply say that $f_n = f_{n-1} + f_{n-2}$, or do you say that the $n$th Fibonacci number is $\frac{\varphi ^n - \psi ^n}{\sqrt{5}}$. In essence, how would you interpret $f_{\pi}$ or the like – Brevan Ellefsen Jan 25 '16 at 5:46
• @BrevanEllefsen thank you for spotting a mistake. My interpretation of fibonacci sequence is $f_n = f_{n-1} + f_{n-2)$ but it would be nice to hear the answer for both cases. – Salvador Dali Jan 25 '16 at 5:57
• Hmm... the problem here is that $f(1.5)$ is outside of the convergence radius... For such a point, not only do you have to determine the meaning of a generating function of a sequence of integers for a non-integer value, but you also have to regularize the sum to have some form of meaning (since it diverges... in the same way we can show that the sum of natural numbers is $-\frac{1}{12}$ in some ways) – Brevan Ellefsen Jan 25 '16 at 6:10
• @BrevanEllefsen sorry, that 1.5 was from old question, when I incorrectly calculated the radius. – Salvador Dali Jan 25 '16 at 6:13
• Here is a quote from wikipedia that might help here: "Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x, and which has the formal power series as its Taylor series; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal power series are not required to give a convergent series when a nonzero numeric value is substituted for x. – Brevan Ellefsen Jan 25 '16 at 6:27