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Do generalized hypergeometric functions $${}_p F_q(a_1,\ldots,a_p; b_1, \ldots,b_q; z) $$ with $p = q+1$ always possess a singularity at $z=1$, independent of the their parameters $a_1,\ldots,a_p$ and $b_1,\ldots,b_q$ under the provision that all cases of finite polynomials are excluded? Can someone mention a reliable source for such a statement or provide a counterexample?

Thanks in advance.

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Voting to close because this thread does not need to keep being bumped by the bot. – anon Oct 21 '10 at 23:13
Why "does not need"? Apparently my answer was not sufficiently useful to OP, and thus deserves a better answer... – J. M. Oct 22 '10 at 0:56

For the $p=q+1$ case, taking the branch cut from 1 to ∞ is conventional so that you can speak of the principal branch of ${}_{q+1} F_q$ ; the function is multivalued outside the unit disk, and 1 and ∞ are branch points.

For unit argument, the key word "Saalschützian" is what you might take into account; see this article by Bühring for more information.

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