# §18.16 Zeros

See §18.2(vi).

## §18.16(ii) Jacobi

Let $\theta_{n,m}$, $m=1,2,\dots,n$, denote the zeros of $P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)$ 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), 18.16 and 18

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 $J_{\alpha}\left(x\right)$10.21(i)). Then

 18.16.6 $\displaystyle\theta_{n,m}$ $\displaystyle\leq\frac{j_{\alpha,m}}{\left(\rho^{2}+\tfrac{1}{12}\left(1-% \alpha^{2}-3\beta^{2}\right)\right)^{\frac{1}{2}}},$ $\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]$, ⓘ Symbols: $[\NVar{a},\NVar{b}]$: closed interval, $\in$: element of, $m$: nonnegative integer, $n$: nonnegative integer, $\rho$ and $\theta_{n,m}$: zeros Referenced by: §18.16(ii) Permalink: http://dlmf.nist.gov/18.16.E6 Encodings: TeX, pMML, png See also: Annotations for 18.16(ii), 18.16(ii), 18.16 and 18 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 $\theta_{n,m}=\phi_{m}+\left(\left(\alpha^{2}-\tfrac{1}{4}\right)\frac{1-\phi_{% m}\cot\phi_{m}}{2\phi_{m}}-\tfrac{1}{4}(\alpha^{2}-\beta^{2})\tan\left(\tfrac{% 1}{2}\phi_{m}\right)\right)\frac{1}{\rho^{2}}+\phi_{m}^{2}O\left(\frac{1}{\rho% ^{3}}\right),$

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 $L^{(\alpha)}_{n}\left(x\right)$ 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), 18.16 and 18

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(iv), 18.16 and 18
 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), 18.16(iv), 18.16 and 18

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(iv), 18.16 and 18
 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), 18.16(iv), 18.16 and 18

See Driver and Jordaan (2013).

### Asymptotic Behavior

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

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

where $a_{m}$ is the $m$th negative zero of $\mathrm{Ai}\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 $H_{n}\left(x\right)$ 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), 18.16 and 18

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 $\mathrm{Ai}\left(x\right)$9.9(i)), $\epsilon_{n,m}<0$, and as $n\to\infty$ with $m$ fixed

 18.16.18 $\epsilon_{n,m}=O\left(n^{-\frac{5}{6}}\right).$ ⓘ Defines: $\epsilon_{n,m}$ (locally) Symbols: $O\left(\NVar{x}\right)$: 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), 18.16 and 18

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 $L^{(\pm\frac{1}{2})}_{n}\left(x\right)$ lead immediately to results for the zeros of $H_{n}\left(x\right)$.

## §18.16(vi) Additional References

For further information on the zeros of the classical orthogonal polynomials, see Szegő (1975, Chapter VI), Erdélyi et al. (1953b, §§10.16 and 10.17), Gatteschi (1987, 2002), López and Temme (1999a), and Temme (1990a).