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15 Hypergeometric Function Notation 15 Hypergeometric Function 15.2 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.
$k,\ell,m,n$	integers.
$s$	nonnegative integer.
$\delta$	arbitrary small positive constant.
$\Gamma\left(z\right)$	gamma function (§5.2(i)).
$\psi\left(z\right)$	$\ifrac{\Gamma'\left(z\right)}{\Gamma\left(z\right)}$ .

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

We use the following notations for the hypergeometric function:

15.1.1		${{}_{2}F_{1}}\left(a,b;c;z\right)=F\left(a,b;c;z\right)=F\left({a,b\atop c};z% \right),$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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		$\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}\left(a,b;c;z% \right)=\mathbf{F}\left({a,b\atop c};z\right)={{}_{2}{\mathbf{F}}_{1}}\left(a,% b;c;z\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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 Function 15.2 Definitions and Analytical Properties

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