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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 $F(-1/12, 5/12, 1, {64/614125})$?

glebovg
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    My guess would be no, since $$(a){n}=(a)(a+1)\cdots(a+n-1)$$ and $$(-a){n}=(-a)(-a+1)\cdots(-a+n-1)=(-1)^{n}(a)(a-1)\cdots(a-n+1)=(-1)^{n}a^{(n)}$$ are very different beasts. Here, $(a)_{n}$ is the rising factorial (a.k.a. Pochhammer symbol) and $a^{(n)}$ is the falling factorial. – parsiad Jul 13 '17 at 21:23
  • I guess no also. You asked about $F(-1/2, 1/2, 1, 1/2)$ which Mathematica evaluates to $\sqrt{2}\mathrm {EllipticE}[-1]/\pi$ and I don't that can be expressed with $\Gamma$ functions. – Somos Jul 14 '17 at 01:10
  • In your example, those $F$ values are $\frac{2}{\pi}K$ and $\frac{2}{\pi}E$ for $k^2=k'^2=\frac{1}{2}$. Therefore $K=K'$, $E=E'$. Now use Legendre's $KE'+K'E-KK'=\frac{\pi}{2}$. Here that works, but that's a very special case of course. – ccorn Jul 15 '17 at 13:02
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    This is a particular case of your previous question. These hypergeometric series are essentially complete elliptic integrals and my answer (https://math.stackexchange.com/a/2354635/72031) shows that both $K, E$ can be expressed in terms of Gamma function and $\pi$ for any singular modulus $k$. The hypergeometric function at the end of the answer is related to Ramanujan's theory of theta functions to alternative bases (Berndt calls this theories with signature $r$, classical case being $r=2$, and your hypergeometric geometric function belongs to $r=6$). The answer is yes for this case also. – Paramanand Singh Jul 17 '17 at 07:49
  • A minor correction in my previous comment : in the theory of signature $6$ the function ${}{2}F{1}(1/6,5/6,1,x)$ plays the role of complete elliptic integral $K$ and the role of $E$ is played by ${}{2}F{1}(-5/6,5/6,1,x)$ and it is this function which can be evaluated in terms of Gamma and $\pi$ for singular moduli $k$ (or $x$). I doubt there is a connection between this function and $ F(-1/12,5/12,1,x)$ of your question. – Paramanand Singh Jul 17 '17 at 17:27

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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.

Cye Waldman
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