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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!

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  • $\begingroup$ You could try the series given here. $\endgroup$
    – Jeremy Rouse
    Commented Jul 26, 2014 at 20:58
  • $\begingroup$ I've seen that one before, it doesn't seem very fast converging, also it requires a lot of calculations per step, am I perhaps missing something? $\endgroup$
    – eatscrayons
    Commented Jul 27, 2014 at 3:22
  • $\begingroup$ Where are you in the complex plane, and how much precision do you want? There is a paper of Schmelzer and Trefethen that discusses computation of $\Gamma$, they give different methods and seem to favor Hankel contours and numerical integration in general. people.maths.ox.ac.uk/trefethen/publication/PDF/2007_122.pdf $\endgroup$
    – NAME_IN_CAPS
    Commented Jul 27, 2014 at 11:11
  • $\begingroup$ If you mean where the function works it is exact for all real and imaginary values for gamma(x). I am unfortunately unfamiliar with much of what is in that paper (although I went through it all). Perhaps some background would be helpful, I am a junior in the Engineering program at my school. I apologize for my lack of proper vernacular for mathematics although there are currently 8 fast converging infinite series for the reciprocal gamma function in my notes. The Professors on campus have been helpful (one recommended using this site) but I do not know what (if anything) to do next. $\endgroup$
    – eatscrayons
    Commented Jul 27, 2014 at 16:03
  • $\begingroup$ How fast does your method compute $\Gamma(1/2)=\sqrt{\pi}$? That would be a useful trial case. Similarly see the formulas for $\Gamma(1/4)$ at en.wikipedia.org/wiki/Particular_values_of_the_Gamma_function And note there that Borwein/Zucker have quadratically fast algorithms for $\Gamma(n/24)$. $\endgroup$
    – NAME_IN_CAPS
    Commented Jul 27, 2014 at 18:31

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