I very well know that the Binomial Theorem can be extended from an integer power, to a rational exponent, any real number and perhaps to the complex field with $z^w=e^{w \cdot ln(z)}$ or $(x+iy)^{u+iv}=\bigl((x+iy)^{iv}\bigr)^u$. Is this theorem extendable to any other field $\mathbb F $ ?
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$\begingroup$ What is your version of the binomial theorem for when the exponent is say $\pi$ ? $\endgroup$– coffeemathMay 7, 2021 at 18:13
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$\begingroup$ Have you heard of the generalized binomial theorem for real exponent ? $\endgroup$– user65203May 7, 2021 at 18:16
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$\begingroup$ For example: $(1+x)^\pi=\sum_{k=0}^\infty \frac{\pi\cdot(\pi-1)\cdot(\pi-2)\cdot\cdot\cdot(\pi-k)}{k!}x^k$ $\endgroup$– Elkin MontoyaMay 7, 2021 at 18:18
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$\begingroup$ Does your formula hold for $x=0,1,2,..$? $\endgroup$– Michael MorrowMay 7, 2021 at 18:51
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1$\begingroup$ @Michael Morrow Now that you mention it, I rechecked the formula and I mistakenly offsetted the coefficients by 1, the product should be until $(a-k+1)$. It should also be mentioned that this particular example only converges for $|x|<1$. c.f. <en.wikipedia.org/wiki/…> $\endgroup$– Elkin MontoyaMay 7, 2021 at 21:04
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