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We've identified a bug in our hypergeometric function, where the approximation that we're using fails.

Basically, we approximate hyp2f1(a, b, c, x) where b==c and |x| < 1.0 as (1.0 - x)^(-a). However, this approximation seems to fail when b is a negative integer such that |b| < 100 (I'm sorry, but I haven't used the math formatting before).

What is a good approximation for the hypergeometric function when b==c and |b| < 100 and b < 0 and b is an integer and |x| < 1.0? Thanks for your help in advance.

Whole Number Fraction

xaav
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  • Actually, do you have a reference for the big-F notation? I haven't seen that before. – xaav Oct 04 '17 at 19:46
  • @RobertIsrael That is not correct. I typed plot 2f1(2, 1, 1, x) and (1.0 - x)^(-2) for x = 0 to 5 into wolframalpha and the results were different from plot 2f1(2, -1, -1, x) and (1.0 - x)^(-2) for x = 0 to 5. Something is going on here. – xaav Oct 04 '17 at 20:10
  • See e.g. MathWorld. – Robert Israel Oct 04 '17 at 20:34

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The formal series for ${}_2F_1([a,b],[c],x)$ is $$ \sum_{k=0}^\infty \frac{(a)_k (b)_k}{k! (c)_k} x^k $$ where $(\cdot)_k$ is a Pochhammer symbol. In the case $b = c$, the factors $(b)_k$ and $(c)_k$ should cancel. However, there is an exception when $b=c$ is a nonpositive integer: $(b)_k = 0$ when $k > -b$, so the coefficient has the indeterminate form $0/0$. The conventional interpretation is that when $b$ is a nonpositive integer, we ignore all terms for $k > -b$. Thus in this case

$$ {}_2F_1([a,b],[b],x) = \sum_{k=0}^{-b} \frac{(a)_k}{k!} x^k $$

For example, your ${}_2F_1([2,-1],[-1],x) = 2x+1$.

Robert Israel
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