# Sum over k of r by r+kt to the Power of k-1 over k Factorial by Power of z

## Theorem

Let $x$ be the continuous real function of $z$ which satisfies:

$\ln x = z x^t$

where $x = 1$ when $z = 0$.

Then:

 $\displaystyle x^r$ $=$ $\displaystyle \sum_{k \mathop \ge 0} \dfrac {r \left({r + k t}\right)^{k - 1} } {k!} z^k$ $\displaystyle$ $=$ $\displaystyle 1 + r z + \dfrac {r \left({r + 2 t}\right)} 2 z^2 + \cdots$