Newtons 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  fracrkrk  factorials are defined for ALL complex numbers except for negative integers  I dont 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 noninteger factorials