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Numerical calculation of a double sum (Appell's function F4)
Hello, I am working currently with double hypergeometric sums. Unfortunately the one I need (Appell's F4) is not implemented in M. So I decided to write my own F4. I discovered quickly that this a far more complicated task than expected, which is due to many special cases of the arguments (e.g. the series can be truncated, transform into another series or have singularities in Gamma functions; there are also cases where these singularities cancel). I managed to overcome all those difficulties, but still I wonder, if there is an easier solution to the following: F4 is a double sum. My approach is a simple While loop (ok, it's more complicated, because cancelling singularities can lead to intermediate terms that are zero and so you have t think carefully about getting a quantity that tells you when the desired accuracy is reached). Is there a function in M that can do double sums numerically and also checks for convergence? I appreciate any help, because this - I would guess - would make the calculation faster and the code more readable. Thanks for any suggestions Markus Huber PS: Don't be misled by the seeming simplicity of Appell's Function F4 when you look it up. In all probability you will only find the standard series representation. There are also other representations with other regions of convergence that exist of 5 single hypergeometric series. Esp. those I need.