DLMF: GA.1 Notation

§GA.1. Notation

$j,m,n$	nonnegative integers.
$k$	nonnegative integer, except in §GA.20.
$x,y$	real variables.
$z=x+iy$	complex variable.
$a,b,q,s,w$	real or complex variables with $\|q\|<1$ .
$\delta$	arbitrary small positive constant.
$\Complex$	complex plane (excluding infinity).
$\Real$	real line (excluding infinity).
$\binom{n}{m}$	binomial coefficient $\frac{n!}{m!(n-m)!}$ .
empty sums	zero.
empty products	unity.

The main functions treated in this chapter are the gamma function $\Gamma\!\left(z\right)$ , the psi function $\psi\!\left(z\right)$ , the beta function $\mathrm{B}\!\left(a,b\right)$ , and the $q$ -gamma function $\Gamma _{{q}}\!\left(z\right)$ .

The notation $\Gamma\!\left(z\right)$ is due to Legendre. Alternative notations for this function are: $\Pi(z-1)$ (Gauss) and $(z-1)!$ . Alternative notations for the psi function are:

$\Psi(z-1)$	Gauss; Jahnke and Emde(1945),
$\psi(z)$	Whittaker and Watson(1927),
$\Psi(z)$	Davis(1933),
$\mathsf{F}(z-1)$	Pairman(1919).