How to solve an equation involving gamma and incomplete gamma function

Question

How can I solve for $t$ in $$S = 1 - \gamma(\alpha/p,(\lambda t)^p) / \Gamma (\alpha / p)$$ where $$ \gamma(s,x)=\int_0^x t^{s-1}e^{-t} dt$$

$S$ is the survival function for a generalized gamma distribution. Thank you in advance!

Answer 1

Define the upper incomplete Gamma function as $$\Gamma(a,z)=\int_z^\infty t^{a-1}\mathrm e^{-t}\mathrm dt \\ =\Gamma(a)-\gamma(a,z)$$ Hence $$S=1-\frac{\Gamma(a/p)-\Gamma\big(a/p,(\lambda t)^p\big)}{\Gamma(a/p)}$$ Define the regularized incomplete Gamma function, $$Q(a,z)=\frac{\Gamma(a,z)}{\Gamma(a)}$$ So $$S=Q\big(a/p,(\lambda t)^p\big)$$ Define the inverse w.r.t $z$, $$Q^{-1}(a,s)~~\text{is the solution for }z~\text{in} \\ s=Q(a,z)$$ Therefore $$Q^{-1}(a/p,S)=(\lambda t)^p \\ \frac{\big(Q^{-1}(a/p,S)\big)^{1/p}}{\lambda}=t$$ $Q^{-1}$ is implemented in the Wolfram Language as InverseGammaRegularized.