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

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) / 4π has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that Γ( 1/4), π, and eπ are algebraically independent.

where δ is the Masser–Gramain constant , although numerical work by Melquiond et al. indicates that this conjecture is false.[1]

Borwein and Zucker have found that Γ( 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 Γ( 1/5) or other denominators.

In particular, where AGM() is the arithmetic–geometric mean, we have[2]

where A is the Glaisher–Kinkelin constant and G is Catalan's constant.

and many more relations for Γ( n/d) where the denominator d divides 24 or 60.[5]

Gamma quotients with algebraic values must be "poised" in the sense that the sum of arguments is the same (modulo 1) for the denominator and the numerator.

Integrating the reciprocal gamma function along the positive real axis also gives the Fransén–Robinson constant.

On the negative real axis, the first local maxima and minima (zeros of the digamma function) are: