# §14.16 Zeros

## §14.16(i) Notation

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

 14.16.1 $\displaystyle\mu$ $\displaystyle=m+\delta_{\mu},$ $\displaystyle\nu$ $\displaystyle=n+\delta_{\nu},$

where $m$, $n\in\Integer$ and $\delta_{\mu}$, $\delta_{\nu}\in(0,1)$. For all cases concerning $\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ and $\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ we assume that $\nu\geq-\frac{1}{2}$ without loss of generality (see (14.9.5) and (14.9.11)).

## §14.16(ii) Interval $-1

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

• (a)

$\mu\leq 0$.

• (b)

$\mu>0$, $n\geq m$, and $\delta_{\nu}>\delta_{\mu}$.

• (c)

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

• (d)

$\nu=0,1,2,3,\dots$.

The number of zeros of $\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ 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 $\delta_{\nu}\leq\delta_{\mu}$.

• (b)

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

The zeros of $\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ in the interval $(-1,1)$ interlace those of $\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$. $\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ 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 $\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)$ and $\mathop{\cos\/}\nolimits\!\left(\mu\pi\right)$ have opposite signs or the same sign. In the special case $\mu=0$ and $\nu=n=0,1,2,3,\dots$, $\mathop{\mathsf{Q}_{n}\/}\nolimits\!\left(x\right)$ has $n+1$ zeros in the interval $-1.

For uniform asymptotic approximations for the zeros of $\mathop{\mathsf{P}^{-m}_{n}\/}\nolimits\!\left(x\right)$ in the interval $-1 when $n\to\infty$ with $m$ $(\geq 0)$ fixed, see Olver (1997b, p. 469).

## §14.16(iii) Interval $1

$\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ has exactly one zero in the interval $(1,\infty)$ if either of the following sets of conditions holds:

• (a)

$\mu>0$, $\mu>\nu$, $\mu\notin\Integer$, and $\mathop{\sin\/}\nolimits\!\left((\mu-\nu)\pi\right)$ and $\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)$ have opposite signs.

• (b)

$\mu\leq\nu$, $\mu\notin\Integer$, and $\lfloor\mu\rfloor$ is odd.

For all other values of $\mu$ and $\nu$ (with $\nu\geq-\frac{1}{2}$) $\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ has no zeros in the interval $(1,\infty)$.

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