Gamma function identities

There are many identities relating the values of the gamma function at one point to values at other points. These identities mostly derive from four basic identities. There are analogous identities for functions related to the gamma function — log gamma, digamma, trigamma, etc. — that follow the same pattern.

The diagram below helps to visualize the identity landscape.

Chart of arguments for gamma function identities

The conjugation identities all have the same form. They say conjugation commutes with function application.
The addition identities relate the values of a function at z and z+n where n is an integer.
The reflection identities relate values of a function at z and 1-z.
The multiplication identities relate the value of a function at integer multiples of z to a sum or product of function applications at fractional shifts of z.

Table of Identities

The following table gives links to the specifics of each kind of identity.

Gamma	Log gamma	Polygamma
conjugation	conjugation	conjugation
addition	addition	addition
reflection	reflection	reflection
multiplication	multiplication	multiplication

The digamma function ψ(z) is the derivative of log Γ(z). The polygamma function ψ⁽ⁿ⁾(z) is the n^th derivative of ψ(z) for integer n ≥ 1 and ψ⁽⁰⁾(z) = ψ(z).

References to A&S are formula numbers in Abramowitz and Stegun.

Gamma identities

Conjugation

A&S 6.1.23

Table

Addition

The simplest case of the addition identity corresponds to n = 1.

A&S 6.1.15

Here is the general case for positive integer n.

A&S 6.1.16

Table

Reflection

A&S 6.1.17

Table

Multiplication

The simplest case of the multiplication identity is the duplication formula.

A&S 6.1.18

Here is the general case for positive integer n.

A&S 6.1.20

Table