Newton's generalized binomial theorem 

Isaac Newton generalized the formula to other exponents by considering an infinite series:

where r can be any complex number (in particular r can be any real number, not necessarily positive and not necessarily an integer), and the coefficients are given by 

This is the same as  \frac{r!}{k!\,(r-k)!}  factorials are defined for ALL complex numbers, except for negative integers

I don't know why you like to remove relevant information, maybe because you like (I do not know for what reason whatsoever) to deny the definition of non-integer factorials!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!