I am looking for some source of summation formulas for the $q$-hypergeometric function ${~}_3\phi_1$ in the sense of Gasper-Rahman book. For some reason, the book focuses heavily on ${~}_{r+1}\phi_r$ case, and I am not sure what's a good way to reduce to this case. To be more precise, there is a specific identity that involves ${~}_3\phi_1$ in base $q$ on one side and ${~}_2\phi_1$ in base $q^2$ on the other side that I am trying to find a reference for (it is messy, but I am perfectly happy to email it to an interested party). I suspect that there is a reduction formula somewhere, but I am just a novice in the area.
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asked Jan 26, 2015 at 18:34
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$\begingroup$ I don't think you will find a better reference than Gasper and Rahman. If it's a ${}_3\phi_1$ I assume it's terminating. There is a transformation from ${}_3\phi_1$ to ${}_2\phi_1$ in the appendices of GR (I don't have access to my copy at the moment). Then you will have ${}_2\phi_1$:s on both sides of the identity, which should help. $\endgroup$– Hjalmar RosengrenCommented Jan 27, 2015 at 4:32
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$\begingroup$ The equation I meant is (III.8) in Gasper and Rahman. $\endgroup$– Hjalmar RosengrenCommented Jan 27, 2015 at 7:25
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$\begingroup$ Thanks! I think I made a bit of a mistake in the calculations and am now debugging. But III.8 looks very relevant indeed. $\endgroup$– Lev BorisovCommented Jan 27, 2015 at 15:04
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$\begingroup$ @HjalmarRosengren I sent you an email with the actual identity (it turns out to be ${~}_3\phi_2$ on the other side). Hope you got it. $\endgroup$– Lev BorisovCommented Jan 29, 2015 at 14:01
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$\begingroup$ Yes, I got the email but I have been busy with other things. I should have time to look at it tomorrow afternoon (Friday). $\endgroup$– Hjalmar RosengrenCommented Jan 29, 2015 at 16:50
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