English Wikipedia @ Freddythechick:Reference desk/Archives/Mathematics/2013 September 12

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September 12

Speculations and Suspicions relating to the Factorial or Gamma function

Any input is welcome ! ( Yes, I am well aware that the statement below hasn't even been proven for all rationals in particular, let alone for all algebraics in general, but at this point I welcome any educated guesses, gut instincts, and informed opinions, or perhaps even possible counter examples I may not yet be aware of ).

$\Gamma (\mathbb {A} \smallsetminus \mathbb {N} )\ \subset \ \mathbb {T} \qquad ;\qquad \Gamma ^{^{-1}}{\Big (}\mathbb {A} \smallsetminus \Gamma (\mathbb {N} ){\Big )}\ \subset \ \mathbb {T}$

where $\mathbb {A}$ and $\mathbb {T}$ stand for algebraics and transcendental numbers, respectively. — 79.113.213.211 (talk) 15:47, 12 September 2013 (UTC)

I suggest you find a math-related forum somewhere else. This is a reference desk where we provide sourced answers to specific questions. We don't do "educated guesses, gut instincts, [or] informed opinions". Rojomoke (talk) 12:18, 13 September 2013 (UTC)

I am not necessarily suggesting that Wikipedia's "virtual librarians" which usually provide answers on this page should do the research and speculating themselves(!) —though frankly I wouldn't mind that either, since they are quite intelligent people— but perhaps someone might be able to bring to my attention objective information concerning (unproven) conjectures and/or (ongoing) mathematical research of other serious mathematicians... which I honestly don't think would go against the Wiki-guide-lines you mentioned... and the answer right below I think is proof of that. — 79.113.226.249 (talk) 06:10, 14 September 2013 (UTC)

From out article Gamma Function: " It has been proved that Γ(n+r) is a transcendental number and algebraically independent of π for any integer n and each of the fractions r = 1/6, 1/4, 1/3, 2/3, 3/4, and 5/6.". From the citation that gives that result, [1], and our page on periods, Ring of periods: Γ(p/q)^q is a period for natural numbers p and q. This paper, which appears to be from 2007, [2] on transcendental values of the Digamma function, the Logarithmic derivative of the Gamma function, indicates that the first part of your problem is still open, and is a long standing open problem. This excerpt, [3], mentions a general point regarding Hilbert's seventh problem, namely the about the expectation that Transcendental functions should take transcendental values at irrational algebraic arguments- the paper itself may be of interest to you. The article Particular values of the Gamma function may be of service to you also.Phoenixia1177 (talk) 13:30, 13 September 2013 (UTC)

Generally speaking,

${\tfrac {1}{n}}!\ =\int _{0}^{\infty }{e^{-x^{n}}}\ dx$

so the transcendence of fractional factorials seems rather straightforward... — 79.113.226.249 (talk) 06:10, 14 September 2013 (UTC)

Also, from the reflection formula

$\left(-{\tfrac {k}{n}}\right)!\ {\tfrac {k}{n}}!\ =\ {\frac {{\tfrac {k}{n}}\cdot \pi }{\sin \left({\tfrac {k}{n}}\cdot \pi \right)}}\ \in \ \mathbb {T}$

it would logically follow that for all rational ⁠k/n⁠ at least one of the two numbers Γ(1 ± ⁠k/n⁠) is transcendental, since their product obviously is, because Sin(⁠k/n⁠ π) is algebraic for all rational ⁠k/n⁠ — See [4] [5] — 79.113.226.249 (talk) 10:15, 14 September 2013 (UTC)

Your observation is accurate, and not without interest; however, it lies in relation to the result, "gamma is transcendental for all noninteger rationals", in the same way that the observation "for all a there is a coprime b so ax + b is prime infinitely often" relates to Dirichlet's theorem. I think a very different approach will be required for the general case- the main meat of the conjecture would seem to be in dealing with gamma(t) for t in (0, 1/2].Phoenixia1177 (talk) 18:18, 14 September 2013 (UTC)

Perhaps combining it with Theorem 12 on page 6 (numbered 440), from the document you offered, might be a start... Actually, the fact that it is already known for a fact that that integral is transcendental was somewhat surprising, since its value represents the area of any given geometric curve described by the binomial equation xⁿ + y^m = 1... which was pretty much what triggered my initial curiosity about the nature of the various values of the factorial or gamma function all along... Now all I need is to know whether the arc length is also transcendental:

$\int _{0}^{1}{{\sqrt {1+\left({\frac {d}{dx}}{\sqrt[{m}]{1-x^{n}}}\right)^{2}}}\ dx}\ \in \ \mathbb {T} \ ?\ ,\ \forall \ m,\ n\ \in \ \mathbb {N} >1$

— 79.113.226.248 (talk) 19:02, 14 September 2013 (UTC)

I'll be honest, this is out of my usual areas of interest (so I don't know much about it), so this will probably be my last contribution (it has been fun, though). But relating to B(a,b) being transcendental- if you could show that {B(1/r, s/r) : s = 1,...r-2};{c = pi/sin(pi/r)} is algebraically independent, then gamma(1/r) ^ r = cB(1/r,1/r)...B(1/r,(r-2)/r) should be transcendental. But, like I said, I'm out of my element; so I seriously doubt there is any real merit to that approach. At any rate, good luck, update me on my talk page if you get any interesting results:-)Phoenixia1177 (talk) 20:35, 14 September 2013 (UTC)

Your contributions have been very helpful. Thank you. — 79.113.226.248 (talk) 20:49, 14 September 2013 (UTC)