Numerical calculation of a double sum (Appell's function F4)

• To: mathgroup at smc.vnet.net
• Subject: [mg75335] Numerical calculation of a double sum (Appell's function F4)
• From: Markus Huber <mhla at gmx.at>
• Date: Wed, 25 Apr 2007 05:42:00 -0400 (EDT)

```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.

```

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