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

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Theorem

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

$\ln x = z x^t$

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


Then:

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


Proof


Sources