15 Hypergeometric FunctionNotation15 Hypergeometric Function15.2 Definitions and Analytical Properties

§15.1 Special Notation

ⓘ

Keywords:

Olver’s, Olver’s hypergeometric function, hypergeometric function, notation

Referenced by:

§13.10(ii), §13.16(i), §13.23(i), §13.4(i), §13.4(ii), §14.3(i), §18.17(vii), §18.26(iv), §18.33(iv), §19.5, §33.6

Permalink:

http://dlmf.nist.gov/15.1

See also:

Annotations for Ch.15

(For other notation see Notation for the Special Functions.)

$x$	real variable.
$z=x+\mathrm{i}y$	complex variable.
$a,b,c$	real or complex parameters.
$k,\mathrm{\ell },m,n$	integers.
$s$	nonnegative integer.
$\delta$	arbitrary small positive constant.
$\mathrm{\Gamma }\left(z\right)$	gamma function (§5.2(i)).
$\psi \left(z\right)$	${\mathrm{\Gamma }}^{\prime }\left(z\right)/\mathrm{\Gamma }\left(z\right)$.

Unless indicated otherwise primes denote derivatives with respect to the variable.

We use the following notations for the hypergeometric function:

15.1.1		$${}_2{}^{}F_1^{}(a,b;c;z)=F(a,b;c;z)=F(\genfrac{}{}{0pt}{}{a,b}{c};z),$$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter Permalink: http://dlmf.nist.gov/15.1.E1 Encodings: TeX, pMML, png See also: Annotations for §15.1 and Ch.15

and also

15.1.2		$$\frac{F(a,b;c;z)}{\mathrm{\Gamma }\left(c\right)}=𝐅(a,b;c;z)=𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)={}_2{}^{}𝐅_1^{}(a,b;c;z),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter Permalink: http://dlmf.nist.gov/15.1.E2 Encodings: TeX, pMML, png See also: Annotations for §15.1 and Ch.15

(Olver (1997b, Chapter 5)).

15 Hypergeometric Function15.2 Definitions and Analytical Properties

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