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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?

Salvador Dali
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    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 05:46
  • @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 05:57
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    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 06:10
  • @BrevanEllefsen sorry, that 1.5 was from old question, when I incorrectly calculated the radius. – Salvador Dali Jan 25 '16 at 06:13
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    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 06:27
  • Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal power series; negative and fractional powers of x are examples of this." – Brevan Ellefsen Jan 25 '16 at 06:27
  • Remember that a generating function is simply a function that has coefficients matching those of a series. The function doesn't have to have anything to do with the series, although it often does. With the Fibonacci sequence, the generating function is useful in finding the closed form of the $n$th term I wrote above – Brevan Ellefsen Jan 25 '16 at 06:31

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I'm quite, quite far from an expert but I do have a specific literature reference to contribute: the whole section 2.4 Power series, analytic theory in the book generatingfunctionology by Herbert S. Wilf. The second edition (1994) is downloadable (there is a third edition but not on the web) and this is how the author introduces the topic in the section:

If they do converge though, and they represent functions, that's a big advantage, for then we may be in a position to find analytic information about the recurrence relation that might not otherwise be easily obtainable.

I think the relevant message is: If you have the skills, it's possible to learn something about the coefficients of the series from the function represented by the power series. If the power series represents a function. But see Wilf for the actual wisdom.

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