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In DLMF 16.4.11: $_3F_2$ $${{}_{3}F_{2}}\left({a,b,c\atop d,e};1\right)=\frac{\Gamma\left(e\right)\Gamma% \left(d+e-a-b-c\right)}{\Gamma\left(e-a\right)\Gamma\left(d+e-b-c\right)}{{}_{% 3}F_{2}}\left({a,d-b,d-c\atop d,d+e-b-c};1\right)$$

For values $a=b=c,d=2 a,e=a+1,$ I don't get agreement for an individual term; say $a=5,k=3$ where $k$ is the usual indexing term for term order.
I do get agreement when $d=a+1,e=2 a$ ; i.e. $d$ and $e$, are swapped.
I thought I would check on a more public forum (here) before submitting a request to NIST/DLMF. I suppose that there might be a global correction factor I am missing; i.e. corrections that apply later; but given the nature of the Generalized Hypergeometric function I doubt it. DLMF does say, "A detailed treatment of analytic continuation in (16.4.11) and asymptotic approximations as the variables approach infinity is given by Aomoto (1987)."
I don't have access to that. But analytic continuation would imply an overlap between domains with restrictive ranges? I am not skilled with analytic continuation though.
This arose from analyzing: Generalized hypergeometric function at unity.
I have maxima code but I thought an outside review would be more objective; and probably neater.

Somos
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rrogers
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    DMLF formula is correct. – Claude Leibovici Sep 10 '21 at 05:19
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    The formula does not claim that the series expansions on each side will agree term-by-term. It claims that the value of each side agrees. You can have two different expansions for the same thing. – Gary Sep 10 '21 at 11:20
  • @ClaudeLeibovici -- can you give an accessible reference? I do have several references, but can't remember/find this relationship. I can't afford Aomoto until I can find it used. I believe you, but I would like to see the proof and reasoning; so I don't ask a dumb question again. – rrogers Sep 10 '21 at 17:17
  • @rrogers.Gary gave the right explanation – Claude Leibovici Sep 10 '21 at 18:05

1 Answers1

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I have found two references proving the statement. One using algebra, the other Calculus. Both are understandable at undergraduate level; I think.
1)Algebraic: Special Functions Section 3.3 Corollary 3.3.5, but the harder, core, part is Theorem 3.3.3; Unfortunately, not available online. Not too bad in price, though; $28 USD used. Theorem 3.3.3 uses two specialized techniques/properties; refactoring (to misuse a software term) of Pochhammer convolutions, and a limit of Pochhammer expressions using the ratio of Gamma functions. One technique I have used as a "trick" and the other new to me.
The difference between a trick and technique is whether you can reuse it.
2)Calculus: A NEW PROOF OF THE FUNDAMENTAL TWO-TERM TRANSFORMATION FOR THE SERIES 3F2(1) DUE TO THOMAE. I explored this line, but took a "wrong turn" and didn't exploit the technique as I could have.
If somebody wants, I will reconstruct 1 along my own eccentric lines; but adhering to the book. They do slip an assumption/restriction in the middle that seems out of place.
Be aware that my retelling will be fairly long.
Looking at the references, this relationship goes back to 1836 :) So there are certainly alternate proofs.

rrogers
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