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

Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise.

The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions).

Among other notations commonly used in the literature Erdélyi et al. (1953a) and Olver (1997b) denote ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mu }\left(x\right)$ by ${\mathrm{P}}_{\nu }^{\mu }(x)$ and ${\mathrm{Q}}_{\nu }^{\mu }(x)$, respectively. Magnus et al. (1966) denotes ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$, $P_{\nu }^{\mu }\left(z\right)$, and $Q_{\nu }^{\mu }\left(z\right)$ by $P_{\nu }^{\mu }(x)$, $Q_{\nu }^{\mu }(x)$, ${𝔓}_{\nu }^{\mu }(z)$, and ${𝔔}_{\nu }^{\mu }(z)$, respectively. Hobson (1931) denotes both ${𝖯}_{\nu }^{\mu }\left(x\right)$ and $P_{\nu }^{\mu }\left(x\right)$ by $P_{\nu }^{\mu }\left(x\right)$; similarly for ${𝖰}_{\nu }^{\mu }\left(x\right)$ and $Q_{\nu }^{\mu }\left(x\right)$.