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5 Gamma FunctionNotation

§5.1 Special Notation

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

j,m,n nonnegative integers.
k nonnegative integer, except in §5.20.
x,y real variables.
z=x+iy complex variable.
a,b,q,s,w real or complex variables with |q|<1.
δ arbitrary small positive constant.
γ Euler’s constant (§5.2(ii)).
primes derivatives with respect to the variable.

The main functions treated in this chapter are the gamma function Γ(z), the psi function (or digamma function) ψ(z), the beta function B(a,b), and the q-gamma function Γq(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) Davis (1933); F(z-1) Pairman (1919).