Extended Binomial Fractional Powers

I'm trying to understand how the binomial theorem can be applied to expand polynomials to fractional powers. I can sought of piece it together without using the theorem, but when I try to use the theorem I run into problems with fractional factorials.

For example how would I expand $\displaystyle (x+y)^{1/2}$ using the binomial theorem?

Re: Extended Binomial Fractional Powers

Re: Extended Binomial Fractional Powers

the way you're going to find:

$\displaystyle \begin{pmatrix}1/2\\k\end{pmatrix}$

is to subtract 0, then 1, then...up to k-1 from 1/2, multiply these all together, and then divide by k! so

$\displaystyle \begin{pmatrix}1/2\\0\end{pmatrix}=1$, by definition (we have no terms).

$\displaystyle \begin{pmatrix}1/2\\1\end{pmatrix}=\frac{1/2}{1!} = 1/2$.

$\displaystyle \begin{pmatrix}1/2\\2\end{pmatrix}=\frac{(1/2)(1/2 - 1)}{2!} = -1/8$.

$\displaystyle \begin{pmatrix}1/2\\3\end{pmatrix}=\frac{(1/2)(1/2 - 1)(1/2 - 2)}{3!} = 1/16$.

after awhile, using the definition directly gets computationally intense, but:

$\displaystyle \begin{pmatrix}1/2\\k\end{pmatrix}=\frac{(1/2)(1/2 - 1)(1/2 - 2)\dots(1/2-k+1)}{k!}$

usually, only the first few terms in the infinite series you get are actually written out (unless your exponent is a non-negative integer, in which case you actually get a finite series).