§15.1 Special Notation

Keywords:: Olver’s hypergeometric function, hypergeometric function
Referenced by:: §13.10(ii), §13.16(i), §13.23(i), §13.4(i), §13.4(ii), ¶ ‣ §14.3(i), ¶ ‣ §18.12, ¶ ‣ §18.17(vii), §18.26(iv), ¶ ‣ §18.33(iv), §18.35(i), §19.5, §33.6
Permalink:: http://dlmf.nist.gov/15.1

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

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

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

We use the following notations for the hypergeometric function:

15.1.1 $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(a,b;c;z\right)=\mathop{F\/}\nolimits\!\left(a,b;c;z\right)=\mathop{F\/}\nolimits\!\left({a,b\atop c};z\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized 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

and also

15.1.2 $\frac{\mathop{F\/}\nolimits\!\left(a,b;c;z\right)}{\mathop{\Gamma\/}\nolimits\!\left(c\right)}=\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)=\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop c};z\right)=\mathop{{{}_{{2}}{\mathbf{F}}_{{1}}}\/}\nolimits\!\left(a,b;c;z\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{{{}_{{p}}{\mathbf{F}}_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : 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

(Olver (1997b, Chapter 5)).