§14.16 Zeros

Throughout this section we assume that $\mu$ and $\nu$ are real, and when they are not integers we write

where $m$, $n\in \mathbb{Z}$ and ${\delta }_{\mu }$, ${\delta }_{\nu }\in (0,1)$. For all cases concerning ${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$ and $P_{\nu }^{\mu }\left(x\right)$ we assume that $\nu \ge -\frac{1}{2}$ without loss of generality (see (14.9.5) and (14.9.11)).

The number of zeros of ${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ is $\max (\lceil \nu -|\mu |\rceil ,0)$ if any of the following sets of conditions hold:

• (a)

  $\mu \le 0$.

• (b)

  $\mu >0$, $n\ge m$, and ${\delta }_{\nu }>{\delta }_{\mu }$.

• (c)

  $\mu >0$, , and $m-n$ is odd.

• (d)

  $\nu =0,1,2,3,\mathrm{\dots }$.

The number of zeros of ${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ is $\max (\lceil \nu -|\mu |\rceil ,0)+1$ if either of the following sets of conditions holds:

• (a)

  $\mu >0$, $n>m$, and ${\delta }_{\nu }\le {\delta }_{\mu }$.

• (b)

  $\mu >0$, , and $m-n$ is even.

The zeros of ${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ interlace those of ${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$. ${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)$ has $\max (\lceil \nu -|\mu |\rceil ,0)+k$ zeros in the interval $(-1,1)$, where $k$ can take one of the values $-1$, $0$, $1$, $2$, subject to $\max (\lceil \nu -|\mu |\rceil ,0)+k$ being even or odd according as $\cos \left(\nu \pi \right)$ and $\cos \left(\mu \pi \right)$ have opposite signs or the same sign. In the special case $\mu =0$ and $\nu =n=0,1,2,3,\mathrm{\dots }$, ${\mathsf{Q}}_n\left(x\right)$ has $n+1$ zeros in the interval .

For uniform asymptotic approximations for the zeros of ${\mathsf{P}}_n^{-m}\left(x\right)$ in the interval when $n\to \mathrm{\infty }$ with $m$ $(\ge 0)$ fixed, see Olver (1997b, p. 469).

$P_{\nu }^{\mu }\left(x\right)$ has exactly one zero in the interval $(1,\mathrm{\infty })$ if either of the following sets of conditions holds:

• (a)

  $\mu >0$, $\mu >\nu$, $\mu \notin \mathbb{Z}$, and $\sin \left((\mu -\nu )\pi \right)$ and $\sin \left(\mu \pi \right)$ have opposite signs.

• (b)

  $\mu \le \nu$, $\mu \notin \mathbb{Z}$, and $\lfloor \mu \rfloor$ is odd.

For all other values of $\mu$ and $\nu$ (with $\nu \ge -\frac{1}{2}$) $P_{\nu }^{\mu }\left(x\right)$ has no zeros in the interval $(1,\mathrm{\infty })$.

${\mathit{Q}}_{\nu }^{\mu }\left(x\right)$ has no zeros in the interval $(1,\mathrm{\infty })$ when $\nu >-1$, and at most one zero in the interval $(1,\mathrm{\infty })$ when .