I am wondering whether it is correct to say, or whether there is a way to prove that, when $a$ is a positive rational number but not an integer, the complete gamma function of $a$, namely $\Gamma(a)$, is always irrational.

In general, I am not sure how to approach a problem involving the rationality of a number if it is defined by integrals.

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    $\begingroup$ Read here: en.wikipedia.org/wiki/Particular_values_of_the_gamma_function Note that in many cases the number $\;\pi\;$ plays a central role. $\endgroup$ – DonAntonio Apr 14 '18 at 6:58
  • $\begingroup$ Thank you. That's why I have the conjecture that they should be irrational in general. $\endgroup$ – Weijun Zhou Apr 14 '18 at 7:02
  • $\begingroup$ @We ...and unless you can prove something specific for any rational non integer argument for $\;\Gamma\;$, and that doesn't look very easy to achieve, it will remain that: a mere conjecture. $\endgroup$ – DonAntonio Apr 14 '18 at 7:03
  • $\begingroup$ Hmm... I wanted to ask what research has been done on this (or what are some possible ways to approach the problem) but I think that question would be too broad and off topic for this site. $\endgroup$ – Weijun Zhou Apr 14 '18 at 7:07

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