Generating Functions
# Fibonacci Numbers
It might be famous that nth fibonacci number $fib_n$ can be solved with complex analysis.
we get nth fibonnaci number just by dumping
$$f(z) = \frac{-z}{z^2 + z - 1}$$
to the residue calculus.
( We might get $fib_n = \frac{1}{\alpha^n(\alpha-\beta)} + \frac{1}{\beta^n(\beta-\alpha)}$ where $\alpha, \beta$ are zeros of $z^2 + z - 1$ . )
# Catalan numbers ?
It there a way of solving
$$c(z) = \frac{1 - \sqrt{1-4z}}{2z}$$
and getting the nth catalan number $catal_n$ in complex analysis?
I might get the instruction from
[this stackexchange](https://math.stackexchange.com/questions/3052399/extract-catalan-numbers-from-generating-function-via-residues). (Excuse me for the interruption.)
# Bernoulli Numbers?
I found its generating function $$b(z) = \frac{1}{z} \frac{d^2}{d(\frac{1}{z})^2} ln \Gamma(\frac{1}{z})$$ and nth Bernoulli Number $$ber_n = - n \zeta(1-n)$$ in
[this wikipedia](https://en.wikipedia.org/wiki/Bernoulli_number#Generating_function)