# §18.16 Zeros

See §18.2(vi).

## §18.16(ii) Jacobi

Let $\theta_{n,m}$, $m=1,2,\dots,n$, denote the zeros of $\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits% \theta\right)$ with

 18.16.1 $0<\theta_{n,1}<\theta_{n,2}<\cdots<\theta_{n,n}<\pi.$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $\theta_{n,m}$: zeros Permalink: http://dlmf.nist.gov/18.16.E1 Encodings: TeX, pMML, png See also: Annotations for 18.16(ii)

Then $\theta_{n,m}$ is strictly increasing in $\alpha$ and strictly decreasing in $\beta$; furthermore, if $\alpha=\beta$, then $\theta_{n,m}$ is strictly increasing in $\alpha$.

### Inequalities

 18.16.2 $\frac{(m-\tfrac{1}{2})\pi}{n+\tfrac{1}{2}}\leq\theta_{n,m}\leq\frac{m\pi}{n+% \tfrac{1}{2}},$ $\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]$,
 18.16.3 $\frac{(m-\tfrac{1}{2})\pi}{n}\leq\theta_{n,m}\leq\frac{m\pi}{n+1},$ $\alpha=\beta$, $\alpha\in[-\tfrac{1}{2},\tfrac{1}{2}]$, $m=1,2,\dots,\left\lfloor\frac{1}{2}n\right\rfloor$.

Also, with $\rho$ defined as in (18.15.5)

 18.16.4 ${\frac{\left(m+\tfrac{1}{2}(\alpha+\beta-1)\right)\pi}{\rho}<\theta_{n,m}<% \frac{m\pi}{\rho}},$ $\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]$,

except when $\alpha^{2}=\beta^{2}=\tfrac{1}{4}$.

 18.16.5 $\theta_{n,m}>\frac{\left(m+\tfrac{1}{2}\alpha-\tfrac{1}{4}\right){\pi}}{n+% \alpha+\tfrac{1}{2}},$ $\alpha=\beta$, $\alpha\in(-\tfrac{1}{2},\tfrac{1}{2})$, $m=1,2,\dots,\left\lfloor\tfrac{1}{2}n\right\rfloor$.

Let $j_{\alpha,m}$ be the $m$th positive zero of the Bessel function $\mathop{J_{\alpha}\/}\nolimits\!\left(x\right)$10.21(i)). Then

 18.16.6 $\displaystyle\theta_{n,m}$ $\displaystyle\leq\frac{j_{\alpha,m}}{\left(\rho^{2}+\tfrac{1}{12}\left(1-% \alpha^{2}-3\beta^{2}\right)\right)^{\frac{1}{2}}},$ $\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]$, 18.16.7 $\displaystyle\theta_{n,m}$ $\displaystyle\geq\frac{j_{\alpha,m}}{\left(\rho^{2}+\tfrac{1}{4}-\tfrac{1}{2}(% \alpha^{2}+\beta^{2})-\pi^{-2}(1-4\alpha^{2})\right)^{\frac{1}{2}}},$ $\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]$, $m=1,2,\dots,\left\lfloor\tfrac{1}{2}n\right\rfloor$.

### Asymptotic Behavior

Let $\phi_{m}=\ifrac{j_{\alpha,m}}{\rho}$. Then as $n\to\infty$, with $\alpha$ ($>-\tfrac{1}{2}$) and $\beta$ ($\geq-1-\alpha$) fixed,

 18.16.8 $\theta_{n,m}=\phi_{m}+\left(\left(\alpha^{2}-\tfrac{1}{4}\right)\frac{1-\phi_{% m}\mathop{\cot\/}\nolimits\phi_{m}}{2\phi_{m}}-\tfrac{1}{4}(\alpha^{2}-\beta^{% 2})\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2}\phi_{m}\right)\right)\frac{1}{% \rho^{2}}+\phi_{m}^{2}\mathop{O\/}\nolimits\!\left(\frac{1}{\rho^{3}}\right),$

uniformly for $m=1,2,\dots,\left\lfloor cn\right\rfloor$, where $c$ is an arbitrary constant such that $0.

### Other Bounds

See Dimitrov and Nikolov (2010), and Driver and Jordaan (2013).

## §18.16(iii) Ultraspherical and Legendre

For ultraspherical and Legendre polynomials, set $\alpha=\beta$ and $\alpha=\beta=0$, respectively, in the results given in §18.16(ii).

## §18.16(iv) Laguerre

The zeros of $\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(x\right)$ are denoted by $x_{n,m}$, $m=1,2,\dots,n$, with

 18.16.9 $0 Symbols: $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.16.E9 Encodings: TeX, pMML, png See also: Annotations for 18.16(iv)

Also, $\nu$ is again defined by (18.15.17).

### Inequalities

For $m=1,2,\dots,n$, and with $j_{\alpha,m}$ as in §18.16(ii),

 18.16.10 $x_{n,m}>\ifrac{j_{\alpha,m}^{2}}{\nu},$ Symbols: $m$: nonnegative integer, $n$: nonnegative integer, $\nu$ and $x$: real variable Referenced by: §18.16(iv), §18.16(iv) Permalink: http://dlmf.nist.gov/18.16.E10 Encodings: TeX, pMML, png See also: Annotations for 18.16(iv)
 18.16.11 $x_{n,m}<(4m+2\alpha+2)\left(2m+\alpha+1+\left((2m+\alpha+1)^{2}+\tfrac{1}{4}-% \alpha^{2}\right)^{\frac{1}{2}}\right)\Big{/}\nu.$ Symbols: $m$: nonnegative integer, $n$: nonnegative integer, $\nu$ and $x$: real variable Referenced by: §18.16(iv) Permalink: http://dlmf.nist.gov/18.16.E11 Encodings: TeX, pMML, png See also: Annotations for 18.16(iv)

The constant $j_{\alpha,m}^{2}$ in (18.16.10) is the best possible since the ratio of the two sides of this inequality tends to 1 as $n\to\infty$.

For the smallest and largest zeros we have

 18.16.12 $x_{n,1}\geq\frac{2n^{2}+\alpha n-n+2\alpha+2-2(n-1)\sqrt{n^{2}+(n+2)(\alpha+1)% }}{n+2},$ Symbols: $n$: nonnegative integer and $x$: real variable Referenced by: §18.16(iv), Other Changes Permalink: http://dlmf.nist.gov/18.16.E12 Encodings: TeX, pMML, png Addition (effective with 1.0.5): This bound, taken from Dimitrov and Nikolov (2010), replaces the weaker bound $x_{n,1}>2n+\alpha-2-(1+4(n-1)(n+\alpha-1))^{\frac{1}{2}}$, which had been taken from Ismail and Li (1992). Reported 2012-07-30 See also: Annotations for 18.16(iv)
 18.16.13 $x_{n,n}\leq\frac{2n^{2}+\alpha n-n+2\alpha+2+2(n-1)\sqrt{n^{2}+(n+2)(\alpha+1)% }}{n+2}.$ Symbols: $n$: nonnegative integer and $x$: real variable Referenced by: §18.16(iv), Other Changes Permalink: http://dlmf.nist.gov/18.16.E13 Encodings: TeX, pMML, png Errata (effective with 1.0.5): This bound, taken from Dimitrov and Nikolov (2010), replaces the weaker bound $x_{n,n}<2n+\alpha-2+(1+4(n-1)(n+\alpha-1))^{\frac{1}{2}}$, which had been taken from Ismail and Li (1992). Reported 2012-07-30 See also: Annotations for 18.16(iv)

See Driver and Jordaan (2013).

### Asymptotic Behavior

As $n\to\infty$, with $\alpha$ and $m$ fixed,

 18.16.14 $x_{n,n-m+1}=\nu+2^{\frac{2}{3}}a_{m}\nu^{\frac{1}{3}}+\tfrac{1}{5}2^{\frac{4}{% 3}}{a_{m}^{2}}\nu^{-\frac{1}{3}}+\mathop{O\/}\nolimits\!\left(n^{-1}\right),$

where $a_{m}$ is the $m$th negative zero of $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$9.9(i)). For three additional terms in this expansion see Gatteschi (2002). Also,

 18.16.15 $x_{n,m}<\nu+2^{\frac{2}{3}}a_{m}\nu^{\frac{1}{3}}+2^{-\frac{2}{3}}{a_{m}^{2}}% \nu^{-\frac{1}{3}},$

when $\alpha\notin(-\frac{1}{2},\frac{1}{2})$.

## §18.16(v) Hermite

All zeros of $\mathop{H_{n}\/}\nolimits\!\left(x\right)$ lie in the open interval $(-\sqrt{2n+1},\sqrt{2n+1})$. In view of the reflection formula, given in Table 18.6.1, we may consider just the positive zeros $x_{n,m}$, $m=1,2,\dots,\left\lfloor\tfrac{1}{2}n\right\rfloor$. Arrange them in decreasing order:

 18.16.16 $(2n+1)^{\frac{1}{2}}>x_{n,1}>x_{n,2}>\cdots>x_{n,\left\lfloor n/2\right\rfloor% }>0.$ Symbols: $\left\lfloor\NVar{x}\right\rfloor$: floor of $x$, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.16.E16 Encodings: TeX, pMML, png See also: Annotations for 18.16(v)

Then

 18.16.17 $x_{n,m}=(2n+1)^{\frac{1}{2}}+2^{-\frac{1}{3}}(2n+1)^{-\frac{1}{6}}a_{m}+% \epsilon_{n,m},$

where $a_{m}$ is the $m$th negative zero of $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$9.9(i)), $\epsilon_{n,m}<0$, and as $n\to\infty$ with $m$ fixed

 18.16.18 $\epsilon_{n,m}=\mathop{O\/}\nolimits\!\left(n^{-\frac{5}{6}}\right).$ Defines: $\epsilon_{n,m}$ (locally) Symbols: $\mathop{O\/}\nolimits\!\left(\NVar{x}\right)$: order not exceeding, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.16.E18 Encodings: TeX, pMML, png See also: Annotations for 18.16(v)

For an asymptotic expansion of $x_{n,m}$ as $n\to\infty$ that applies uniformly for $m=1,2,\dots,\left\lfloor\tfrac{1}{2}n\right\rfloor$, see Olver (1959, §14(i)). In the notation of this reference $x_{n,m}=u_{a,m}$, $\mu=\sqrt{2n+1}$, and $\alpha=\mu^{-\frac{4}{3}}a_{m}$. For an error bound for the first approximation yielded by this expansion see Olver (1997b, p. 408).

Lastly, in view of (18.7.19) and (18.7.20), results for the zeros of $\mathop{L^{(\pm\frac{1}{2})}_{n}\/}\nolimits\!\left(x\right)$ lead immediately to results for the zeros of $\mathop{H_{n}\/}\nolimits\!\left(x\right)$.