I was researching the Gamma function and came across this in Wikipedia:
The positive solution to Γ(z − 1) = Γ(z + 1) is z = φ ≈ +1.618, the golden ratio, and the common value is Γ(φ − 1) = Γ(φ + 1) = φ! ≈ +1.44922960226989660037.
This is blowing my mind. My mind was blown learning $e^{iπ} = -1$ and the link between the Riemann Zeta Function and prime numbers. There is some sort of link between factorials and the golden ratio? Not only is φ a solution to the equation, but φ! is the value of each side?!?
The citation link goes to the list of integer sequences at OEIS, which doesn't give any indication (that I can find) as to why.
What is the link between the the Golden Ratio and the Gamma function?