Axiom Documentation

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)