Particular values of the gamma function: Difference between revisions

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where {{math|''θ''₁}} and {{math|''θ''₄}} are two of the [[Theta function|Jacobi theta functions]].

Certain values of the gamma function can also be written in terms of the [[hypergeometric function]]. For instance, <math>\Gamma\left(\frac{1}{4}\right)^{4}=\frac{32\pi^{3}}{\sqrt{33}} {}_{3}F_{2}\left(\frac{1}{2},\ \frac{1}{6},\ \frac{5}{6};\ 1,\ 1;\ \frac{8}{1331}\right)</math>

and

<math>\Gamma\left(\frac{1}{3}\right)^{6}=\frac{12\pi^{4}}{\sqrt{10}} {}_{3}F_{2}\left(\frac{1}{2},\ \frac{1}{6},\ \frac{5}{6};\ 1,\ 1;\ -\frac{9}{64000}\right)</math>

however it is an open question whether this is possible for all rational inputs to the gamma function. <ref>Johansson, F. (2023). Arbitrary-precision computation of the gamma function. ''Maple Transactions'', ''3''(1). <nowiki>https://doi.org/10.5206/mt.v3i1.14591</nowiki></ref>

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