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.

  • $\begingroup$ Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. $\endgroup$ – José Carlos Santos May 12 at 7:48
  • $\begingroup$ 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. $\endgroup$ – Start wearing purple May 14 at 15:35
  • $\begingroup$ Thanks very much for the information! $\endgroup$ – 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 limitationenter image description here

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.

  • $\begingroup$ Thank you so much! $\endgroup$ – adixit May 19 at 5:34

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