${}_2F_1$ as a FINITE series: How is this result obtained?

Question

I am using the following result:

which I have found in this link:

http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/03/06/07/10/0001

I am trying to find out how this result has been obtained (analytically).

I guess it comes from using the Eurler's transformation here:

http://en.wikipedia.org/wiki/Hypergeometric_function#Fractional_linear_transformations

and doing something else but I cannot find how the infinite sum that defines ${}_2F_1$ can be simplified to a finite sum as in the result shown above.

Any ideas? Thank you in advance!

Answer 1

Well I have been investigating and that result simply comes from subsituting the parameters' values in the following expression:

_{(source: wolfram.com)}

After some algebraic manipulations the result is obtained.

The finite series comes from the fact that $(2-n)_k$ is zero for $k > n-2$, which allows the truncation of the series.