Let $F(a, b, c, x)$ be the hypergeometric function. Suppose I can express $F(a, b, c, x)$ in finite terms, say in terms of $\Gamma$ functions, for various $x$. Is there a way I can then deduce the value of $F(-a, b, c, x)$? For example, it is known that $$F\bigg(\frac{1}{2}, \frac{1}{2}, 1, \frac{1}{2}\bigg) = \frac{\Gamma(1/4)^2}{2\pi^{3/2}}.$$ Is there a way to use this result to deduce the value of $F(-1/2, 1/2, 1, 1/2)$? Likewise, it is known that $$F\left(\frac{1}{12}, \frac{5}{12}, 1, \Big(\frac{4}{85}\Big)^3\right) = \frac{\Gamma(1/7) \Gamma(2/7) \Gamma(4/7)}{8\pi^2} \frac{255^{1/4}}{7^{1/4}}.$$ Is there a way to use this result to deduce the value of $F(-1/12, 5/12, 1, {64/614125})$?
2026-03-27 07:11:42.1774595502
Relationship between two hypergeometric functions
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In general, when you are given solutions for special values of $z$ in $\text{F}(a,b,c,z)$ they are unique, unless otherwise specified. I have found three solutions in terms of gamma functions for $z=1/2$ here; none of them would allow you to find the specific answer that you seek.
As for your second case, I don't imagine it would be any different. However, I cannot say for certain as I am unable find anything like that. I did, however, ascertain that it is correct insofar as I was able to verify it numerically and from the gamma solution.