Hypergeometric Functions Identities: n_F_n & (n+1)_F_n

[benorin]

TL;DR Summary

Two multiple integral identities for n_F_n and (n+1)_F_n from my paper (unpublished) I'm curious if they are new as I've never seen them before but I'm not well read on p_F_q so I'm hoping one of you can give me a reference

See attachment for identities and proofs, if you find my proofs are incorrect in some way please post it. Thanks for your time.

 

[jasonRF]

If I read your notes correctly, theorem 4.2 is
$_{n+1} F_n\left(\begin{array}\\ a_1, \ldots, a_{n+1} \\ b_1, \ldots, b_n \end{array}; z \right) = \Xi_n$
and the proof of theorem 4.3 says
$\Xi_n = _n F_n\left(\begin{array}\\ a_1, \ldots, a_n \\ b_1, \ldots, b_n \end{array}; z \right)$
I don't see how that can possibly be true, since it would mean the value of $a_{n+1}$ in $_{n+1}F_n$ is irrelevant. I didn't attempt to work through the math, so I don't know if I just misunderstood something, or there was a typo or what.

jason

 

[benorin]

If I read your notes correctly, theorem 4.2 is
$_{n+1} F_n\left(\begin{array}\\ a_1, \ldots, a_{n+1} \\ b_1, \ldots, b_n \end{array}; z \right) = \Xi_n$
and the proof of theorem 4.3 says
$\Xi_n = _n F_n\left(\begin{array}\\ a_1, \ldots, a_n \\ b_1, \ldots, b_n \end{array}; z \right)$
I don't see how that can possibly be true, since it would mean the value of $a_{n+1}$ in $_{n+1}F_n$ is irrelevant. I didn't attempt to work through the math, so I don't know if I just misunderstood something, or there was a typo or what.

jason

One of your $\Xi_n 's$ is an $\Omega_n$ in the proof I posted. You probably read that wrong, but thanks for somebody finally posting on this, it's sad to compare the views of the post to the views of the PDF imho. Ouchy, an extra click or three... Thank you for taking the time to open the attachment.

 

[Haborix]

They are known. See here.

 

[benorin]

They are known. See here.

I submitted to that site the PDF in the OP and several days later that page appeared. Not going to speculate, maybe they saw my PDF and were like "oh, yeah those identities... let's add them." The second of those was a generalized version of what I had, but I didn't know it before reading that page. This is the email.

 

[Haborix]

The site says they were added in 2001, so I'm not sure what's going on.

 

[benorin]

The site says they were added in 2001, so I'm not sure what's going on.

IDK either, maybe they were there already and I missed them, to be honest I wasn't that hopeful.

 

[jasonRF]

One of your $\Xi_n 's$ is an $\Omega_n$ in the proof I posted. You probably read that wrong, but thanks for somebody finally posting on this, it's sad to compare the views of the post to the views of the PDF imho. Ouchy, an extra click or three... Thank you for taking the time to open the attachment.

Well, in your proof of 4.3 you state that $\Xi_n$ is the RHS of theorem 4.2. So I guess you define both $\Omega_n$ and $\Xi_n$ as the RHS of theorem 2. Seems like a typo.

anyway, I usually do not click on PDFs, either. They can contain malware/viruses, or they indicate something that is too long for the OP to type - either because it is a long question that will take tons of time to read and answer, or because the OP doesn’t care enough to bother typing the question themselves.

 

[benorin]

yes there was a typo in the proof of thm. 4.3 where it reads thm. 4.2 should be thm. 4.3. By the way, I already typed it up using MathType and MS Word and you know how much a headache typing up tex with so many indices and nested operands and stuff, I just printed to PDF and post as an attachment saved me an hour and a half of typing and reformatting that way, I hadn't considered people weren't opening it because of potential malware, thanks for the tip.