The Gauss theorem $${_2F_1}(a,b;c;1)=\frac{\Gamma(c-a)\Gamma(c-b)}{\Gamma(c)\Gamma(c-a-b)}$$ allows to compute the analytic continuation of ${_2F_1}(a,b;c;1)$ for $a+b>c$ when the series definition diverges. The same can be done for $_3F_2(1)$ via Thomae relations. My question is how to find analytic continuation for $_4F_3(1)$ to the subdomains of $\mathbb{C}^7$ where the sum of the upper parameters is greater than the sum of the lower parameters, so that the series diverges. I know that no direct analogues of Thomae relations exit in this case so the formulas may be more complicated. I have seen some work by Allen Miller and other authors giving transformations for $_4F_3(1)$, but these transformations leave the excess (total upper parameters minus total lower parameters) invariant, so that divergent series is transformed into divergent series. The same question, of course, pertains to $_pF_{p-1}(1)$ with $p>4$...

Any help is highly appreciated.