# §14.1 Special Notation

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

 $x$, $y$, $\tau$ real variables. complex variable. unless stated otherwise, nonnegative integers, used for order and degree, respectively. general order and degree, respectively. complex degree, $\tau\in\mathbb{R}$. Euler’s constant (§5.2(ii)). arbitrary small positive constant. logarithmic derivative of gamma function (§5.2(i)). $\ifrac{\mathrm{d}\psi\left(x\right)}{\mathrm{d}x}$ . Olver’s scaled hypergeometric function: $\ifrac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}$.

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

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

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