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

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

The notation $\mathrm{\Gamma }\left(z\right)$ is due to Legendre. Alternative notations for this function are: $\mathrm{\Pi }(z-1)$ (Gauss) and $(z-1)!$. Alternative notations for the psi function are: $\mathrm{\Psi }(z-1)$ (Gauss) Jahnke and Emde (1945); $\mathrm{\Psi }(z)$ Davis (1933); $𝖥(z-1)$ Pairman (1919).