Transformations relating 3F2 at z with 3F2's at 1/z

Question

I am searching for some transformations for a 3F2 hypergeometric function which send the argument z to 1/z. I am aware of the one given in NIST book (p. 410, Formula 16.8.8) in the special case q=2 (with the notations given in there). However, that requires the differences between the top parameters a1, a2, and a3 (taken two at a time) to not be integers. I was wondering if there are transformations in which this condition could be relaxed. I see most of the results involve unit argument which is not what I am seeking. Any reference in this regard will be greatly appreciated. Thank you.

Answer 1

Unfortunately the answer is no. In the case of connection formulas that bring you to $z\rightarrow\frac{1}{z}$ the coefficients $(a_1-a_2),(a_1-a_3),(a_2-a_3)\not\in\mathbb{Z}$ because the $\Gamma$ Euler function is not defined in $\mathbb{Z}^-$. To convince yourself you can see that also for the Gaussian hypergemetric function $_2F_1$ there is a similar limitation

Alternatively you can look (here at pag.190) the transformation formula proposed by Whipple (1927) for $z\rightarrow\frac{-4z}{(1-z)^2}$ and by Bailey (1929)for $z\rightarrow\frac{-27z}{4*(1-z)^3}$, but both only for particular combinations of the Pochhammer coefficients.