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