13 Confluent Hypergeometric Functions Notation 13 Confluent Hypergeometric Functions 13.2 Definitions and Basic Properties

§13.1 Special Notation

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Keywords:: Kummer functions, Olver’s confluent hypergeometric function, Whittaker functions, confluent hypergeometric functions, notation
Referenced by:: §13.6(v), §9.10(v), (9.6.25), (9.6.26), §9.6(iii)
Permalink:: http://dlmf.nist.gov/13.1
See also:: Annotations for Ch.13

(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)).
$\Gamma\left(x\right)$	gamma function (§5.2(i)).
$\psi\left(x\right)$	$\ifrac{\Gamma'\left(x\right)}{\Gamma\left(x\right)}$ .

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

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

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

13 Confluent Hypergeometric Functions 13.2 Definitions and Basic Properties

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