# §5.1 Special Notation

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

$j,m,n$ nonnegative integers. nonnegative integer, except in §5.20. real variables. complex variable. real or complex variables with $|q|<1$. arbitrary small positive constant. Euler’s constant (§5.2(ii)). derivatives with respect to the variable.

The main functions treated in this chapter are the gamma function $\Gamma\left(z\right)$, the psi function (or digamma 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); $\Psi(z)$ Davis (1933); $\mathsf{F}(z-1)$ Pairman (1919).