# OpenMath Content Dictionary: hypergeo1

Canonical URL:
http://www.math.kobe-u.ac.jp/OCD/
CD File:
hypergeo1.ocd
CD as XML Encoded OpenMath:
hypergeo1.omcd
Defines:
hypergeometric0F1, hypergeometric1F1, hypergeometric2F1, hypergeometric_pFq
Date:
2002-11-29
Version:
0 (Revision 1)
Review Date:
2017-12-31
Status:
experimental

Author: Yasushi Tamura

This CD defines the Gauss hypergeometric function, confluent hypergeometric functions, and generalized hypergeometric functions in one variable. 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.

## hypergeometric0F1

Description:

Hypergeometric function {}_0 F_1.

Commented Mathematical property (CMP):
hypergeometric0F1(;a;z) =\sum_{n=0}^{+\infty} \frac{1}{pochhammer(a,n)pochhammer(1,n)} z^n
Formal Mathematical property (FMP):
$\mathrm{hypergeometric0F1}\left(a,z\right)=\sum _{n=0}^{\infty }\frac{\frac{1}{\mathrm{pochhammer}\left(a,n\right)}}{\mathrm{pochhammer}\left(1,n\right)}{z}^{n}$
Signatures:
sts

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## hypergeometric1F1

Description:

Kummer's confluent hypergeometric function.

Commented Mathematical property (CMP):
hypergeometric1F1(a,b;z) =\sum_{n=0}^{+\infty} \frac{pochhammer(a,n)}{pochhammer(1,n)pochhammer(b,n)} z^n
Formal Mathematical property (FMP):
$\mathrm{hypergeometric1F1}\left(a,b,z\right)=\sum _{n=0}^{\infty }\frac{\mathrm{pochhammer}\left(a,n\right)}{\mathrm{pochhammer}\left(b,n\right)\mathrm{pochhammer}\left(1,n\right)}{z}^{n}$
Signatures:
sts

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## hypergeometric2F1

Description:

The Gauss hypergeometric function. This function has a branch cut on [1,+infinity).

Commented Mathematical property (CMP):
hypergeometric2F1(a,b,c;z) =\sum_{n=0}^{+\infty} \frac{pochhammer(a,n)pochhammer(b,n)}{pochhammer(c,n)pochhammer(1,n)} z^n
Formal Mathematical property (FMP):
$\mathrm{hypergeometric2F1}\left(a,b,c,z\right)=\sum _{n=0}^{\infty }\frac{\mathrm{pochhammer}\left(a,n\right)\mathrm{pochhammer}\left(b,n\right)}{\mathrm{pochhammer}\left(c,n\right)\mathrm{pochhammer}\left(1,n\right)}{z}^{n}$
Example:
z (1-z) d^2 F/dz^2 + (c - (a+b+1) z) d F/dz - a b F = 0
$z\left(1-z\right)\frac{d}{dz}\left(\frac{d}{dz}\left(\mathrm{hypergeometric2F1}\left(a,b,c,z\right)\right)\right)+\left(c-a+b+1\right)z\frac{d}{dz}\left(\mathrm{hypergeometric2F1}\left(a,b,c,z\right)\right)-ab\mathrm{hypergeometric2F1}\left(a,b,c,z\right)=0$
Signatures:
sts

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## hypergeometric_pFq

Description:

Generalized hypergeometric function. This function has a branch cut on [1,+infinity).

Commented Mathematical property (CMP):
hypergeometric_pFq(a,b;z) =\sum_{n=0}^{+\infty} \frac{\Pi_i pochhammer(a_i,n)}{\Pi_i pochhammer(b_i,n)pochhammer(1,n)} z^n
Formal Mathematical property (FMP):
$\mathrm{hypergeometric_pFq}\left(a,b,z\right)=\sum _{n=0}^{\infty }\frac{\frac{\prod _{i=1}^{\mathrm{size}\left(a\right)}}{\mathrm{pochhammer}\left(\mathrm{vector_selector}\left(i,a\right),n\right)}}{\mathrm{pochhammer}\left(1,n\right)}{z}^{n}$
Signatures:
sts

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