§13.1 Special Notation

Keywords:: Kummer functions, Olver’s confluent hypergeometric function, Whittaker functions, confluent hypergeometric functions
Referenced by:: §13.6(v), §9.10(v), §9.6(iii), §9.6(iii)
Permalink:: http://dlmf.nist.gov/13.1

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

$m$	integer.
$n$ , $s$	nonnegative integers.
$x$ , $y$	real variables.
$z$	complex variable.
$\delta$	arbitrary small positive constant.
$\gamma$	Euler’s constant (§5.2(ii)).
$\mathop{\Gamma\/}\nolimits\!\left(x\right)$	Gamma function (§5.2(i)).
$\mathop{\psi\/}\nolimits\!\left(x\right)$	$\ifrac{{\mathop{\Gamma\/}\nolimits^{{\prime}}}\!\left(x\right)}{\mathop{\Gamma\/}\nolimits\!\left(x\right)}$ .

The main functions treated in this chapter are the Kummer functions $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ and $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ , Olver’s function $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ , and the Whittaker functions $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ and $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ .

Other notations are: $\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left(a;b;z\right)$ (§16.2(i)) and $\Phi(a;b;z)$ (Humbert (1920)) for $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ ; $\Psi(a;b;z)$ (Erdélyi et al. (1953a, §6.5)) for $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ ; $V(b-a,b,z)$ (Olver (1997b, p. 256)) for $e^{z}\mathop{U\/}\nolimits\!\left(a,b,-z\right)$ ; $\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)\mathscr{M}_{{\kappa,\mu}}$ (Buchholz (1969, p. 12)) for $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ .

For an historical account of notations see Slater (1960, Chapter 1).