§14.2 Differential Equations

Standard solutions: ${𝖯}_{\nu }\left(\pm x\right)$, ${𝖰}_{\nu }\left(\pm x\right)$, ${𝖰}_{-\nu -1}\left(\pm x\right)$, $P_{\nu }\left(\pm x\right)$, $Q_{\nu }\left(\pm x\right)$, $Q_{-\nu -1}\left(\pm x\right)$. ${𝖯}_{\nu }\left(x\right)$ and ${𝖰}_{\nu }\left(x\right)$ are real when $\nu \in ℝ$ and $x\in (-1,1)$, and $P_{\nu }\left(x\right)$ and $Q_{\nu }\left(x\right)$ are real when $\nu \in ℝ$ and $x\in (1,\mathrm{\infty })$.

Standard solutions: ${𝖯}_{\nu }^{\mu }\left(\pm x\right)$, ${𝖯}_{\nu }^{-\mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{\mu }\left(\pm x\right)$, ${𝖰}_{-\nu -1}^{\mu }\left(\pm x\right)$, $P_{\nu }^{\mu }\left(\pm x\right)$, $P_{\nu }^{-\mu }\left(\pm x\right)$, ${𝑸}_{\nu }^{\mu }\left(\pm x\right)$, ${𝑸}_{-\nu -1}^{\mu }\left(\pm x\right)$.

(14.2.2) reduces to (14.2.1) when $\mu =0$. Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations ${𝖯}_{\nu }^0\left(x\right)={𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }^0\left(x\right)={𝖰}_{\nu }\left(x\right)$, $P_{\nu }^0\left(x\right)=P_{\nu }\left(x\right)$, $Q_{\nu }^0\left(x\right)=Q_{\nu }\left(x\right)$, ${𝑸}_{\nu }^0\left(x\right)={𝑸}_{\nu }\left(x\right)=Q_{\nu }\left(x\right)/\mathrm{\Gamma }\left(\nu +1\right)$.

${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, and ${𝖰}_{\nu }^{\mu }\left(x\right)$ are real when $\nu$, $\mu$, and $\tau \in ℝ$, and $x\in (-1,1)$; $P_{\nu }^{\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ are real when $\nu$ and $\mu \in ℝ$, and $x\in (1,\mathrm{\infty })$.

Unless stated otherwise in §§14.2–14.20 it is assumed that the arguments of the functions ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mu }\left(x\right)$ lie in the interval $(-1,1)$, and the arguments of the functions $P_{\nu }^{\mu }\left(x\right)$, $Q_{\nu }^{\mu }\left(x\right)$, and ${𝑸}_{\nu }^{\mu }\left(x\right)$ lie in the interval $(1,\mathrm{\infty })$. For extensions to complex arguments see §§14.21–14.28.

Equation (14.2.2) has regular singularities at $x=1$, $-1$, and $\mathrm{\infty }$, with exponent pairs $\{-\frac{1}{2}\mu ,\frac{1}{2}\mu \}$, $\{-\frac{1}{2}\mu ,\frac{1}{2}\mu \}$, and $\{\nu +1,-\nu \}$, respectively; compare §2.7(i).

When $\mu -\nu \ne 0,-1,-2,\mathrm{\dots }$, and $\mu +\nu \ne -1,-2,-3,\mathrm{\dots }$, ${𝖯}_{\nu }^{-\mu }\left(x\right)$ and ${𝖯}_{\nu }^{-\mu }\left(-x\right)$ are linearly independent, and when $\mathrm{\Re }\mu \ge 0$ they are recessive at $x=1$ and $x=-1$, respectively. Hence they comprise a numerically satisfactory pair of solutions (§2.7(iv)) of (14.2.2) in the interval . When $\mu -\nu =0,-1,-2,\mathrm{\dots }$, or $\mu +\nu =-1,-2,-3,\mathrm{\dots }$, ${𝖯}_{\nu }^{-\mu }\left(x\right)$ and ${𝖯}_{\nu }^{-\mu }\left(-x\right)$ are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair.

When $\mathrm{\Re }\mu \ge 0$ and $\mathrm{\Re }\nu \ge -\frac{1}{2}$, $P_{\nu }^{-\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ are linearly independent, and recessive at $x=1$ and $x=\mathrm{\infty }$, respectively. Hence they comprise a numerically satisfactory pair of solutions of (14.2.2) in the interval . With the same conditions, $P_{\nu }^{-\mu }\left(-x\right)$ and ${𝑸}_{\nu }^{\mu }\left(-x\right)$ comprise a numerically satisfactory pair of solutions in the interval .