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We can calculate the factorials of whole ( positive) numbers, factorials of fractions whose denominators are $2$ i.e. $\left(1/2\right)!$,

  • but can we also calculate $\left(1/3\right)!$, $\left(2/3\right)!$ and $\left(1/6\right)!$ ?.
  • If yes, what are they ?. Do they also contain the square root of $\pi$, or maybe some other cool constant ?. How can we do that ?.
  • I am familiar with $\Gamma\left(x\right)$ and $\Pi\left)(x\right)$, but not with the incomplete gamma function.

If not, why not ?.

Felix Marin
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DefinitelyNotADolphin
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1 Answers1

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To my knowledge, there is no known formula for the gamma function at non integer rationals $\Gamma(\frac pq)$ except for when $q = 2$ (reminder: $x!$ stands for $\Gamma(x+1)$). Euler's reflection formula is a great contributor for giving a formula when $q = 2$ ($\forall x\notin \mathbb Z, \Gamma(x)\Gamma(1-x) = \frac{\pi}{\sin(\pi x)}$, see Can Euler's reflection formula be used to calculate $\Gamma (1/3)$?), but it falls short for greater values of $q$, and every value of $\Gamma(\frac pq)$ with $p \wedge q = 1$ and $q > 2$ has no closed expression yet (see https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function#General_rational_argument).

rafilou2003
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  • @ralifou2003 I just realised this formula is pretty similar to another formula I found while doing some math in desmos. It is the following:

    $$\cos(x)=\frac{\pi}{\left(-\frac{x}{\pi}-\frac{1}{2}\right)! \left(\frac{x}{\pi}-\frac{1}{2}\right)!}$$

    After some simplification I did just now I got:

    $$\sin(x)=\frac{\pi}{\left(-\frac{x}{\pi}\right)! \left(\frac{x}{\pi}-1\right)!}$$ $$\sin(\pi x)=\frac{\pi}{\left(-x\right)! \left(x-1\right)!}$$ $$\left(-x\right)! \left(x-1\right)!=\frac{\pi}{\sin(\pi x)}$$

     – DefinitelyNotADolphin Nov 21 '24 at 17:25
  • @DefinitelyNotADolphin Yes, you graphed exactly Euler's reflection formula. – rafilou2003 Nov 21 '24 at 17:27
  • @ralifou2003 I was trying to fit the nth row of pascals triangle into a graph so that the the middle value was one (by scaling a bit), and after a while I noticed a pattern. So I generalised and found that when $n=-\frac{1}{2}$ it looked like $\cos(\pi x)$, and I filled in that value and did nothing with it for a while as I deemed it useless and way more complicated method of computing a variant of cosine.

    I fantasized that I was the first person to discover it, and that it would be a breakthrough in maths, but independently discovering something is definitely not what I'd ever expect.

     – DefinitelyNotADolphin Nov 21 '24 at 17:34
  • @ralifou2003 this was my work https://www.desmos.com/calculator/ftk3dxyh1m?lang=en – DefinitelyNotADolphin Nov 21 '24 at 17:35
  • $\displaystyle a! = \Gamma\left(a + 1\right)$ – Felix Marin Nov 21 '24 at 20:55