# How do I get other representations of the Gamma function?

In my studymaterial I see different definitions of the Gamma function In Mathematica it is for example:

Gamma[z] == 1/z Product[(1 + 1/k)^z/(1 + z/k),
{k, 1, Infinity}]/; Not[Element[-z, Integers]\[And] -z >= 0]


( z={0,1,2,..} Is there an easy way to get (derive) some other forms of the Gamma function here defined ? Hoping that one of the forms looks the same as in my studymaterial.

Example: $$Pi(s) = \prod _{n=1}^{\infty } \frac{n^{1-s} (n+1)^s}{n+s}$$

• See functions.wolfram.com/GammaBetaErf/Gamma to this end. Commented Feb 19, 2022 at 13:09
• @user64494 , thanks i will look there if i can derive a wanted form? Commented Feb 19, 2022 at 16:04

There is MathematicalFunctionData which can be exploited to get various representations of the Euler Gamma function. As a starting point one can evaluate e.g.

MathematicalFunctionData["Properties"]


to get a perspective what kind of properties one can search for, in case of Gamma it works like e.g.

MathematicalFunctionData["Gamma", "AlternativeRepresentations"]


MathematicalFunctionData["Gamma", "NamedIdentities"][[19 ;; 23]] //


A kind of convenient browser of mathematical properties one can get evaluating

Manipulate[ Entity["MathematicalFunction", "Gamma"][z],
{z, Entity["MathematicalFunction", "Gamma"]["Properties"]}]

• Thanks, getting information in the traditional form can make searching for my formula easier. Commented Feb 19, 2022 at 19:24
• @janhardo You are welcome! You have also link to Wolfram functions MathematicalFunctionData["Gamma", "WolframFunctionsSiteLink"]. Nontheless I find the browser with Manipulate a more convenient way of exploiting built-in mathematical data . Commented Feb 19, 2022 at 19:31

Use Entity["MathematicalFunction", "Gamma"]

(prodRep = #[z] & /@
Most[Entity["MathematicalFunction", "Gamma"][
"ProductRepresentations"]]) // Column[#, Frame -> All] & //

Also look at Entity["MathematicalFunction", "Gamma"]["Properties"]