Wikipedia:Reference desk/Archives/Mathematics/2013 September 12

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

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,, 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, 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, , 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,
 * $$\tfrac1n !\ = \int_0^\infty{e^{-x^n}}\ dx$$
 * 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 &Gamma;(1 ± $k⁄n$) is transcendental, since their product obviously is, because Sin($k⁄n$ &pi;) is algebraic for all rational $k⁄n$ — See  — 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 + ym = 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:
 * $$\int_0^1{\sqrt{1+\left(\frac{d}{dx}\sqrt[m]{1-x^n}\right)^2}\ dx}\ \in\ \mathbb{T}\ ?\ ,\ \forall\ m,\ n\ \in\ \N>1$$
 * — 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)