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For $n \in \mathbb{N}$ and $a,b \in \mathbb{C}$, there is a closed form expansion for the difference of powers

\begin{equation} a^n - b^n = (a - b) \ \sum^{n - 1}_{j = 0} a^{n - j - 1} b^j. \end{equation}

Is there an analogous expansion for the difference of two numbers raised to non-integer powers? Say, letting $s \in \mathbb{R}$, can we write something of the form

\begin{equation} a^s - b^s = (a - b) \ f(a, b, s) \end{equation}

for some function $f$?

NoFishLikeIan
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  • First you'd have to define exactly what you mean by $a^s$ when $a$ is negative or not real. But once that is settled, the answer is an obvious "yes". Just define $$f(a,b,s) = \dfrac{a^s - b^s}{a-b}$$Now if you want $f$ to have some particular form, that would be trickier. But the binomial theorem is actually still true when the power is not integer. It just gives an infinite series as its result. – Paul Sinclair Aug 31 '22 at 20:39

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