# Is the Binomial Theorem extendable to other fields?

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 $$\mathbb F$$ ?

• What is your version of the binomial theorem for when the exponent is say $\pi$ ? May 7, 2021 at 18:13
• Have you heard of the generalized binomial theorem for real exponent ?
– user65203
May 7, 2021 at 18:16
• For example: $(1+x)^\pi=\sum_{k=0}^\infty \frac{\pi\cdot(\pi-1)\cdot(\pi-2)\cdot\cdot\cdot(\pi-k)}{k!}x^k$ May 7, 2021 at 18:18
• Does your formula hold for $x=0,1,2,..$? May 7, 2021 at 18:51
• @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/…> May 7, 2021 at 21:04