The generalized gamma distribution is described as follows $$f(x)=\frac{\gamma \cdot \left(\frac{x-\mu}{\beta} \right)^{\alpha\cdot \gamma-1}}{\beta\cdot\Gamma(\alpha)}\cdot e^{- \left(\frac{x-\mu}{\beta} \right)^ \gamma}$$
How do I get a gamma distribution on the form of third Pearson distribution:
$$f(x)=\frac{a^b}{\Gamma(b)}\cdot (x-c)^{b-1} \cdot e^{- a\cdot (x-c)}$$
and 'simple' gamma distribution when $c=0$:
$$f(x)=\frac{a^b}{\Gamma(b)}\cdot x^{b-1} \cdot e^{- a\cdot x}$$
When I set for example $\beta=1$ and $\mu=0$, $\alpha=b$, $\gamma=a$ I get $$f(x)=\frac{a}{\Gamma(b)}\cdot x^{a\cdot b-1} \cdot e^{- a\cdot x^a}$$ Which I am unable to transform to the form of 'simple' gamma distribution. How to derive one form to another?