§14.1 Special Notation

Keywords:: Ferrers functions, Legendre functions, Legendre functions on the cut, Mehler functions, associated Legendre functions, conical functions, ring functions
Referenced by:: Notation for the Special Functions
Permalink:: http://dlmf.nist.gov/14.1

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

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