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The following ratio:

$$\frac{\Gamma(2/5)^3}{\pi\Gamma(1/5)}$$

has kept appearing in my research, and the only thing I know about its value is that it is $\cong 0.7567213$, whence the following two questions:

Is the value of this ratio an algebraic number?

What is the exact value of this ratio?

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1 Answer 1

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This number is expected to be transcendental. This answer gives a conceptual framework for studying the algebraicity of such $\Gamma$ ratios, and in fact a completely explicit criterion (which is only conjectural, when it comes to establishing transcendence).

Your number is equal to \begin{equation*} \frac{\Gamma(2/5)^3}{\Gamma(1/5)^2 \Gamma(4/5)} \end{equation*} up to multiplication by an algebraic number. This ratio is described by the vector of exponents $(-2,3,0,-1)$ and the criterion there is not met for $u=2$. In fancy terms, the Bernoulli distribution $B_1$ does not vanish on this vector.

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  • $\begingroup$ ...which itself can be rewritten as $~\dfrac{B\bigg(\dfrac25,~\dfrac25\bigg)}{B\bigg(\dfrac15,~\dfrac15\bigg)}~.$ $\endgroup$
    – Lucian
    Commented Jul 19, 2020 at 23:21

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