The good Captain is using the generalised binomial theorem ..... Another link
I just want to add a little bit more. What you have is the finite binomial theorem when the exponent is an integer.
It has the form,
It must be finite because if is a positive integer then eventually among we hit zero, and the series shall terminate.
But if you think of as a non-integer then it will go on forever. And it will be the binomial (infinite) expansion. Which will converge to the function.
That is only for positive powers (after you sort out the powers to give the actual expansion), but the binomial expansion works for any power.
The general form used here is:
Note if is an integer the coefficients are all zero for .
You also have to worry about convergence for non-integer powers.
RonL