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Recall the definition of the hypergeometric function $$_2F_1(a,b,c;z)=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{n!(c )_n}x^n$$ where $(a)_n$ is defined to be $a(a+1)\cdots(a+n-1)$.

We suppose that none of the $a,b,c$ or the difference between any two of them is an integer.

Here, I am interested in the case where $a>c>0,b<0$.

What's the behaviour of this function on $[0,1]$? I used Mathmatica to plot it and it suggests that it always goes negative somewhere on $[0,1]$. As $_2F_1(a,b,c;0)=1$, this suggests that the function has at least a zero on $[0,1]$.

Is there anyway to prove it?

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Hint:

$_2F_1(a,b,c;z)=~_2F_1(b,a,c;z)$

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