There are rapidly converging infinite series for Pi and the such but it is difficult to locate those for either the gamma or reciprocal gamma function. I am searching for rapidly converging series (and algorithms are also welcome as are recursive solutions). The information being requested is required for use as a comparison.

So far some time has been spent with Borwein's gamma function http://imajna.oxfordjournals.org/content/12/4/519

with one specific example being, $\Gamma(1/2)= \frac{1}{\sqrt[4]{2}}\frac{AG[1]}{\sqrt{\frac{1}{\sqrt{2}}\Sigma[1]}}$

By taking advantage of AGM and making use of summations that take the difference of an^2 and bn^2 it would seem this function would be rapidly convergent however it is only good for a countable number of values of gamma without the introduction of some rather profound mathematical gymnastics.

Overall the scope of his algorithm may be limited but for the values it can calculate the decimal approximation for accuracy appears to be astounding and grow radically with each step although full verification is still required in order to verify this claim and to see to what degree and to how many terms this holds true.

The algorithm for this purpose can be simplified by removing the constants leaving,

$\frac{1}{\left(\sum_{n=0}^{\infty} \left((an)^2-(bn)^2\right)\right)^\frac{1}{2}}-\left({\sum_{n=0}^{\infty} (an)^2-(bn)^2}\right)^\frac{1}{2}$

(My apologies for the badly written Latex, I will work on improving it)

The next step is to build a function that shows the rate of convergency for each additional calculated step for the approximation of Borwein's algorithm and that should wrap this portion up (not asking for help, just outlining steps). One method would be to use MatLab but pen and paper are more enjoyable so will likely prevail.

Are there any other resources (e.g. links) to fast converging series for either the gamma or reciprocal gamma function preferably with full functionality for the variable? Recursive identities are welcome as are exceptions for extremely rapidly converging algorithms with limitations.

Thank you all for the helpful comments and suggestions. If you see any errors please point them out immediately!