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Efficient Computation involving a Hypergeometric function...


I have a discrete distribution:

\!\(f\ \  = \ \ Binomial[x + n, x]\ p\^\(x + 1\)\ \((1 - p)\)\^n\)

Here p and x are fixed,  p is real between 0 and 1, x and n are
integers greater than or equal to zero, and n is the parameter
(n,0,Infinity).

The CDF is

\!\(1 - \(\((1 -
     p)\)\^\(1 + n\)\ p\^\(1 + x\)\ Ã[2 + n + x]\ \
Hypergeometric2F1Regularized[1, 2 + n + x, 2 + n, 1 - p]\)\/Ã[1 + x]\)

where n is actually Floor[n].

Does anybody see a simplification or asymtotic for the CDF that
provides an efficient way to compute this for large n (say n  ge
1000)?  I need to solve CDF==k for various fixed p and x.

Thanks in advance!
-- 
Richard Palmer


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