Gamma function, showing identity of terms

Question

How can I show this identity? $$t^{a+h-1}e^{-t}=t^{a-1}e^{-t}\sum_{k = 0}^{\infty} \frac{ \log(t)^k}{k!}h^k $$ Anyone can give me hint?

Answer 1

By definition, $$e^x = \sum_{n = 0}^\infty \frac{x^n}{n!}.$$

Now $t^h = \exp(\log(t^h)) = \exp(h\log t) \ldots$