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. Other fractional arguments can be approximated through efficient infinite products, infinite series, and recurrence relations.

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

and hence

For non-positive integers, the gamma function is not defined.

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

or equivalently, for non-negative integer values of $n$ :

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

and by means of the reflection formula,

In analogy with the half-integer formula,

where $n! (p)$ denotes the $p$ th multifactorial of $n$ . Numerically,

It is unknown whether these constants are transcendental in general, but $Γ(1 / 3)$ and $Γ(1 / 4)$ were shown to be transcendental by G. V. Chudnovsky. $Γ(1 / 4) / 4 \sqrt π$ has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that $Γ(1 / 4)$ , $π$ , and $e π$ are algebraically independent.

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