I am currently studying the periods of the quintic Calabi-Yau. These are given by solutions to PF operators, which can be expressed in terms of hypergeometric functions. By geometric reasons, a certain combination of hypergeometric functions has to vanish, namely
$2\frac{ \, _4F_3\left(\frac{1}{5},\frac{1}{5},\frac{1}{5},\frac{1}{5};\frac{2}{5},\frac{3}{5},\frac{4}{5};1\right)}{\Gamma \left(\frac{4}{5}\right)^5}-10\frac{ \, _4F_3\left(\frac{2}{5},\frac{2}{5},\frac{2}{5},\frac{2}{5};\frac{3}{5},\frac{4}{5},\frac{6}{5};1\right)}{\Gamma \left(\frac{3}{5}\right)^5}+25\frac{ \, _4F_3\left(\frac{3}{5},\frac{3}{5},\frac{3}{5},\frac{3}{5};\frac{4}{5},\frac{6}{5},\frac{7}{5};1\right)}{\Gamma \left(\frac{2}{5}\right)^5}-\frac{125}{3}\frac{ \, _4F_3\left(\frac{4}{5},\frac{4}{5},\frac{4}{5},\frac{4}{5};\frac{6}{5},\frac{7}{5},\frac{8}{5};1\right)}{ \Gamma \left(\frac{1}{5}\right)^5}=0$
Numerically one can check that this is indeed correct. Such points exist not only for the quintic, but any CY 3-fold. Now I am wondering
- Is there a direct way to prove the vanishing of the combination?
- Does an identity exist which generalizes this, i.e. linear combinations of generalized hypergeometric $_4F_3$ functions of the same differential operator that vanish at special rational arguments? This would be useful to understand the general case.