The Fibonacci function and its generating function.

1. Prove that the following equation holds: \(\mathscr F (z) = z + z \mathscr F (z) + z^2 \mathscr F(z)\) where \(\phi = \frac{1+\sqrt{5}}{2}\text{ and }\hat \phi = \frac{1 - \sqrt{5}}{2}\)

   We know that \(F_x = F_{x-1} + F_{x-2}, \forall x \geq 2, x \in N\)

   So:

   \[\begin{aligned} z + z\mathscr F (z) + z^2 \mathscr F(z) &= z + \sum_{i=0}^{\infty} F_i z^{i+1} + \sum_{i=0}^{\infty} F_i z^{i+2} \\ &= z + \sum_{i=1}^{\infty} F_{i-1}z^i + \sum_{i=2}F_{i-2} z^i \\ &= z + F_0 \cdot z + \sum_{i=2}^{\infty} (F_{i-1} + F_{i-2}) z^i \\ &= z + \sum_{i=2}^{\infty} F_i z^i \\ &= \mathscr F(z) \end{aligned}\]

2. Prove that \(\mathscr F (z) = \frac{z}{1 - z - z^2} = \frac{z}{(1 - \phi z)(1 - \hat\phi z)} = \frac{1}{\sqrt{5}} \left( \frac{1}{1 - \phi z} - \frac{1}{1 - \hat\phi z} \right)\)

   By working the equation in problem 1.

3. Prove that \(\mathscr F(z) = \sum_{i = 0}^{\infty} \frac{1}{\sqrt{5}} (\phi^i - \hat{\phi}^i) z\)

   We can derive from \(F_x = F_{x-1} + F_{x-2}\) that

   \[\begin{aligned} F_x + \frac{\sqrt{5}-1}{2} F_{x-1} =& {\left(\frac{1+\sqrt{5}}{2}\right)}^{x-1} \\ F_x - \frac{\sqrt{5}+1}{2} F_{x-1} =& {\left(\frac{1-\sqrt{5}}{2}\right)}^{x-1} \end{aligned}\]

   By working out the above equations we have: \(F_{x} = \frac{(1+\sqrt{5})^x - (1 - \sqrt{5})^x}{2^x\sqrt{5}}\)

4. Convergence Domain

   Since \(\lim_{n \to \infty} \sqrt[n]{F_n} = \frac{1 + \sqrt{5}}{2}\) and \(\lim_{n \to \infty} \mathscr F (\pm \frac{2}{1+\sqrt{5}}) \neq C\)

   The convergence domain is: \(\left(-\frac{2}{1+\sqrt{5}}, \frac{2}{1+\sqrt{5}}\right) \Leftrightarrow \left(-\frac{\sqrt{5}-1}{2}, \frac{\sqrt{5} - 1}{2}\right)\)