# §14.1 Special Notation

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

 $x$, $y$, $\tau$ real variables. complex variable. 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)). $\ifrac{\mathrm{d}\mathop{\psi\/}\nolimits\!\left(x\right)}{\mathrm{d}x}$ . Olver’s scaled hypergeometric function: $\ifrac{\mathop{F\/}\nolimits\!\left(a,b;c;z\right)}{\mathop{\Gamma\/}\nolimits% \!\left(c\right)}$.

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

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

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