User blog comment:Chronolegends/Tetration on ordinals/@comment-1605058-20160524190634/@comment-1605058-20160524205112

Actually, I am fairly sure that there are no examples of $$\lambda\geq\omega,k,n<\omega$$ with $$n>1$$ for which binomial theorem correctly expands $$(\lambda+k)^n$$.

As for transfinite exponents, I highly doubt there is any reasonable analogue of binomial formula, given that it doesn't even hold for finite exponents.