# §14.27 Zeros

$\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\pm i0\right)$ (either side of the cut) has exactly one zero in the interval $(-\infty,-1)$ if either of the following sets of conditions holds:

• (a)

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

• (b)

$\mu,\nu\in\Integer$, $\mu+\nu<0$, and $\nu$ is odd.

For all other values of the parameters $\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\pm i0\right)$ has no zeros in the interval $(-\infty,-1)$.

For complex zeros of $\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(z\right)$ see Hobson (1931, §§233, 234, and 238).