The discussion centers on the identities and proofs related to hypergeometric functions, specifically the theorems 4.2 and 4.3. Theorem 4.2 states that the hypergeometric function ##_{n+1} F_n(a_1, \ldots, a_{n+1}; b_1, \ldots, b_n; z) = \Xi_n##, while theorem 4.3 claims ##\Xi_n = _n F_n(a_1, \ldots, a_n; b_1, \ldots, b_n; z)##. Participants express confusion regarding the relevance of the parameter ##a_{n+1}## in theorem 4.2 and identify a typographical error in the proof of theorem 4.3. The discussion also highlights the challenges of formatting mathematical expressions accurately in digital formats.
PREREQUISITESMathematicians, researchers in mathematical analysis, and students studying hypergeometric functions will benefit from this discussion, particularly those interested in the accuracy of mathematical proofs and typesetting.
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.jasonRF said: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
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 said:They are known. See here.
IDK either, maybe they were there already and I missed them, to be honest I wasn't that hopeful.Haborix said:The site says they were added in 2001, so I'm not sure what's going on.
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.benorin said: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.