§14.16 Zeros

Permalink:: http://dlmf.nist.gov/14.16

§14.16(i) Notation
§14.16(ii) Interval $-1<x<1$
§14.16(iii) Interval $1<x<\infty$

§14.16(i) Notation

Permalink:: http://dlmf.nist.gov/14.16.i

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

14.16.1

$\mu=m+\delta _{{\mu}},$

$\nu=n+\delta _{{\nu}},$

Symbols:: $m$ : nonnegative integer, $n$ : nonnegative integer, $\mu$ : general order, $\nu$ : general degree and $\delta _{\mu}$
Permalink:: http://dlmf.nist.gov/14.16.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

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 $-1<x<1$

Notes:: See Hobson (1931, pp. 386–388 and 399–401).
Keywords:: Ferrers functions
Permalink:: http://dlmf.nist.gov/14.16.ii

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)

$\mu>0$ , $n<m$ , 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)

$\mu>0$ , $n<m$ , 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 $-1<x<1$ .

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

§14.16(iii) Interval $1<x<\infty$

Notes:: See Hobson (1931, pp. 388–389 and 399).
Keywords:: associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.16.iii

$\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ 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 $\nu<-1$ .

§14.16 Zeros

Contents

§14.16(i) Notation

§14.16(ii) Interval

§14.16(iii) Interval

§14.16(ii) Interval $-1<x<1$

§14.16(iii) Interval $1<x<\infty$