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14 Legendre and Related FunctionsNotation

§14.1 Special Notation

(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).

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