The binomial coefficients

where $n$ is a non-negative integer and  $r\in\{0,\,1,\,2,\,\ldots,\,n\}$,  can be generalized for all integer and non-integer values of $n$ by using the reduced form

here $r$ may be any non-negative integer.  Then Newton’s binomial series gets the simple form

It is not hard to show that the radius of convergence of this series is 1.  This series expansion is valid for every complex number $\alpha$ when  $|z|<1$,  and it presents such a branch of the power $(1\!+\!z)^{{\alpha}}$ which gets the value 1 in the point  $z=0$.

In the case that $\alpha$ is a non-negative integer and $r$ is great enough, one factor in the numerator of

vanishes, and hence the corresponding binomial coefficient ${\alpha\choose r}$ equals to zero; accordingly also all following binomial coefficients with a greater $r$ are equal to zero.  It means that the series is left to being a finite sum, which gives the binomial theorem.

For all complex values of $\alpha$, $\beta$ and non-negative integer values of $r$, $s$, the Pascal’s formula

and Vandermonde’s convolution

hold (the latter is proved by expanding the power $(1\!+\!z)^{{\alpha+\beta}}$ to series).  Cf. Pascal’s rule and Vandermonde identity.