Digital Library of Mathematical Functions
The Gamma Function -- Notation

### Notation

j , m , n | nonnegative integers.
k | except in Physical Applications
x , y | real variables.
z = x + ⅈ y | complex variable.
a , b , q , s , w | real or complex variables with ∣ q ∣ < 1 .
δ | arbitrary small positive constant.
ℂ | complex plane (excluding infinity).
ℝ | real line (excluding infinity).
( n m ) | binomial coefficient n ! m ! ( n - m ) ! .
empty sums | zero.
empty products | unity.

The main functions treated in this chapter are the gamma function Γ ( z ) ,the psi function ψ ( z ) ,the beta function B ( a , b ) , and the q -gamma function Γ q ( z ) .

The notation Γ ( z ) is due to Legendre. Alternative notations for this function are: Π ( z - 1 ) (Gauss) and ( z - 1 ) ! . Alternative notations for the psi function are:

Ψ ( z - 1 ) | Gauss; Jahnke and Emde(1945)
--- | ---
Ψ ( z ) | Whittaker and Watson(1927)
Ψ ( z ) | Davis(1933)
F ( z - 1 ) | Pairman(1919)