Particular values of the gamma function

The Gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general.

Integers and half-integers

For positive integer arguments, the Gamma function coincides with the factorial, that is,

$\Gamma (n+1)=n!\quad \quad n\in \mathbb {N}$

and hence

$\Gamma (1)=1\,$

$\Gamma (2)=1\,$

$\Gamma (3)=2\,$

$\Gamma (4)=6\,$

$\Gamma (5)=24.\,$

For positive half-integers, the function values are given exactly by

$\Gamma (n/2)={\sqrt {\pi }}{\frac {(n-2)!!}{2^{(n-1)/2}}},$

or equivalently,

$\Gamma (n+1/2)={\sqrt {\pi }}{\frac {(2n-1)!!}{2^{n}}},$

where n!! denotes the double factorial. In particular,

$\Gamma (1/2)\,$	$={\sqrt {\pi }}\,$	$\approx 1.7724538509055160273\,$
$\Gamma (3/2)\,$	$={\frac {\sqrt {\pi }}{2}}\,$	$\approx 0.8862269254527580137\,$
$\Gamma (5/2)\,$	$={\frac {3{\sqrt {\pi }}}{4}}\,$	$\approx 1.3293403881791370205\,$
$\Gamma (7/2)\,$	$={\frac {15{\sqrt {\pi }}}{8}}\,$	$\approx 3.3233509704478425512\,$

and by means of the reflection formula,

$\Gamma (-1/2)\,$	$=-2{\sqrt {\pi }}\,$	$\approx -3.5449077018110320546\,$
$\Gamma (-3/2)\,$	$={\frac {4{\sqrt {\pi }}}{3}}\,$	$\approx 2.3632718012073547031.\,$

General rational values

In analogy with the half-integer formula,

$\Gamma (n+1/3)=\Gamma (1/3){\frac {(3n-2)!^{3}}{3^{n}}}$

$\Gamma (n+1/4)=\Gamma (1/4){\frac {(4n-3)!^{4}}{4^{n}}}$

$\Gamma (n+1/p)=\Gamma (1/p){\frac {(pn-(p-1))!^{p}}{p^{n}}}$

where $n!^{k}$ denotes the k:th multifactorial of n. Thus, the Gamma function of any rational argument $p/q$ can be expressed in closed algebraic form in terms of $\Gamma (1/q)$. However, no closed expressions are known for the numbers $\Gamma (1/q)$ where q > 2. Numerically,

$\Gamma (1/3)\approx 2.6789385347077476337$

$\Gamma (1/4)\approx 3.6256099082219083119$

$\Gamma (1/5)\approx 4.5908437119988030532$

$\Gamma (1/6)\approx 5.5663160017802352043$

$\Gamma (1/7)\approx 6.5480629402478244377$

It is unknown whether these constants are transcendental in general, but $\Gamma (1/3)$ was shown to be transcendental by Le Lionnais in 1983 and Chudnovsky showed the transcendence of $\Gamma (1/4)$ in 1984. $\Gamma (1/4)/\pi ^{-1/4}$ has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that $\Gamma (1/4),\pi$ and $e^{\pi }$ are algebraically independent.

The number $\Gamma (1/4)$ is related to the lemniscate constant S by

$\Gamma (1/4)={\sqrt {{\sqrt {2\pi }}S}},$

and it has been conjectured that

$\Gamma (1/4)=\left(4\pi ^{3}e^{2\gamma -\mathrm {\rho } +1}\right)^{1/4}$

where ρ is the Masser-Gramain constant.

Borwein and Zucker have found that $\Gamma (n/24)$ can be expressed algebraically in terms of π, $K(k(1))$, $K(k(2))$, $K(k(3))$ and $K(k(6))$ where $K(k(N))$ is a complete elliptic integral of the first kind. This permits efficiently approximating the Gamma function of rational arguments to high precision using quadratically convergent arithmetic-geometric mean iterations. No similar relations are known for $\Gamma (1/5)$ or larger denominators.

References

• Borwein, J. M. & Zucker, I. J. Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind. IMA J. Numerical Analysis 12, 519-526, 1992.
• X. Gourdon & P. Sebah. Introduction to the Gamma Function
• S. Finch. Euler Gamma Function Constants
• Weisstein, Eric W. "Gamma Function". MathWorld.