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

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.

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• If $a_i-a_j\in\mathbb Z$, then $_3F_2(\ldots,1/z)$ are not defined and the basis of solutions at $z=\infty$ has a more involved form (local expansions of solutions also involve logarithmic terms, see for example 15.3.14 in Abramowitz-Stegun in the case of $_2F_1$). I cannot provide a reference for analogous result for $_3F_2$, meanwhile it should be known since the connection formulas between the two bases of solutions can be obtained from the Mellin-Barnes integrals, see e.g. Section 4.2 of pages.uoregon.edu/njp/beukers.pdf for the non-resonant case. – Start wearing purple May 14 at 15:35
• Thanks very much for the information! – adixit May 19 at 5:33

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.