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I want to solve an equation that involves a Gauss hypergeometric function. The function is as follows $$\,_2F_1\left(1,a,b,-pz\right)$$ where $a<1,b<2$, $p>0$ but $p$ does not have a high value (maybe around 1), the value of $z$ is smaller than $1$. Is there any nice looking approximation for the above function and please also specify some range for the use of that approximation in terms of $pz$. Thanks in advance.

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You almost have it from definition $$\,_2F_1\left(1,a,b,x\right)=1+\frac{a}{b}x+\frac{a (a+1) }{b (b+1)}x^2+\frac{a (a+1) (a+2) }{b (b+1) (b+2)}x^3+O\left(x^4\right)$$ You also could use Padé approximants (built around $x=0$) such as $$\frac{1+\frac{ a-b}{b (b+1)}x}{1-\frac{a+1 }{b+1}x}$$ or $$\frac{1+\frac{ (3 a-4 b-ab)}{b (b+3)}x+\frac{2 \left(a^2-2 a b-a+b^2+b\right)}{b (b+1) (b+2) (b+3)} x^2}{1-\frac{2 (a+2) }{b+3}x+\frac{(a+1) (a+2) }{(b+2) (b+3)}x^2}$$

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