# OpenMath Content Dictionary: hypergeo0

Canonical URL:
http://www.math.kobe-u.ac.jp/OCD/
CD File:
hypergeo0.ocd
CD as XML Encoded OpenMath:
hypergeo0.omcd
Defines:
beta, gamma, pochhammer
Date:
2002-11-29
Version:
0 (Revision 1)
Review Date:
2017-12-31
Status:
experimental

  Author: Yasushi Tamura


This CD defines some basic hypergeometric integrals and symbols necessary to define hypergeometric functions. These functions are described in the following books. (1) Handbook of Mathematical Functions, Abramowitz, Stegun (2) Higher transcendental functions. Vol. III. Krieger Publishing Co., Inc., Melbourne, Fla., 1981, Erdlyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (3) From Gauss to Painleve, Vieweg, Katsunori Iwasaki, Hironobu Kimura, Shun Shimomura, Masaaki Yoshida

## gamma

Description:

Euler's gamma function

Commented Mathematical property (CMP):
gamma(z)=\int_0^{+\infty} t^{z-1} e^{-z} dt (Re(z)>0)
Formal Mathematical property (FMP):
$\mathrm{real}\left(z\right)>0⇒\mathrm{gamma}\left(z\right)=\underset{0}{\overset{\infty }{\int }}{t}^{\left(z-1\right)}{e}^{\left(-z\right)}dt$
Example:
gamma(n) = (n-1)! (n \in N)
$n\in \mathbb{N}⇒\mathrm{gamma}\left(n\right)=\left(n-1\right)!$
Signatures:
sts

 [Next: beta] [Last: pochhammer] [Top]

## beta

Description:

Euler's beta function

Commented Mathematical property (CMP):
beta(p,q)=\frac{gamma(p)gamma(q)}{gamma(p+q)}(p,q \not\in Z_{<=0})
Formal Mathematical property (FMP):
$-p\notin \mathbb{N}\wedge -q\notin \mathbb{N}⇒\frac{\mathrm{gamma}\left(p\right)\mathrm{gamma}\left(q\right)}{\mathrm{gamma}\left(p+q\right)}$
Example:
beta(p,q)=\int_0^1 t^{p-1} (1-t)^{q-1} dt (Re(p),Re(q)>0)
$\mathrm{real}\left(p\right)>0\wedge \mathrm{real}\left(q\right)>0⇒\mathrm{beta}\left(p,q\right)=\underset{0}{\overset{1}{\int }}{t}^{\left(p-1\right)}{\left(1-t\right)}^{\left(q-1\right)}dt$
Signatures:
sts

 [Next: pochhammer] [Previous: gamma] [Top]

## pochhammer

Description:

Pochhammer symbol

Commented Mathematical property (CMP):
pochhammer(a,n) = gamma(a+n)/gamma(a)
Formal Mathematical property (FMP):
$\mathrm{pochhammer}\left(\mathrm{alpha},n\right)=\frac{\mathrm{gamma}\left(\mathrm{alpha}+n\right)}{\mathrm{gamma}\left(\mathrm{alpha}\right)}$
Example:
pochhammer(a,n) = \prod_0^{n-1} (a+i)
$\mathrm{pochhammer}\left(a,n\right)=\prod _{i=0}^{n-1}a+i$
Signatures:
sts

 [First: gamma] [Previous: beta] [Top]