14 Legendre and Related Functions Notation 14 Legendre and Related Functions 14.2 Differential Equations

§14.1 Special Notation

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Keywords:: Ferrers functions, Legendre functions, Legendre functions on the cut, Mehler functions, associated Legendre functions, conical functions, degree, notation, order, ring functions
Referenced by:: Notation for the Special Functions
Permalink:: http://dlmf.nist.gov/14.1
See also:: Annotations for Ch.14

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

$x$ , $y$ , $\tau$	real variables.
$z=x+iy$	complex variable.
$m$ , $n$	unless stated otherwise, nonnegative integers, used for order and degree, respectively.
$\mu$ , $\nu$	general order and degree, respectively.
$-\frac{1}{2}+i\tau$	complex degree, $\tau\in\mathbb{R}$ .
$\gamma$	Euler’s constant (§5.2(ii)).
$\delta$	arbitrary small positive constant.
$\psi\left(x\right)$	logarithmic derivative of gamma function (§5.2(i)).
$\psi'\left(x\right)$	$\ifrac{\mathrm{d}\psi\left(x\right)}{\mathrm{d}x}$ .
$\mathbf{F}\left(a,b;c;z\right)$	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 $Q^{\mu}_{\nu}\left(x\right)$ .