# §18.16 Zeros

See §18.2(vi).

## §18.16(ii) Jacobi

Let $\theta_{n,m}=\theta_{n,m}^{(\alpha,\beta)}$, $m=1,2,\dots,n$, denote the zeros of $P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)$ as function of $\theta$ 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 Ch.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 $\theta_{n,m}^{(-\frac{1}{2},\frac{1}{2})}=\frac{(m-\tfrac{1}{2})\pi}{n+\tfrac{% 1}{2}}\leq\theta_{n,m}^{(\alpha,\beta)}\leq\frac{m\pi}{n+\tfrac{1}{2}}=\theta_% {n,m}^{(\frac{1}{2},-\frac{1}{2})},$ $\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]$,
 18.16.3 $\theta_{n,m}^{(-\frac{1}{2},-\frac{1}{2})}=\frac{(m-\tfrac{1}{2})\pi}{n}\leq% \theta_{n,m}^{(\alpha,\alpha)}\leq\frac{m\pi}{n+1}=\theta_{n,m}^{(\frac{1}{2},% \frac{1}{2})},$ $\alpha\in[-\tfrac{1}{2},\tfrac{1}{2}]$, $m=1,2,\dots,\left\lfloor\frac{1}{2}n\right\rfloor$.

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 Ch.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, Legendre and Chebyshev

For ultraspherical and Legendre polynomials, set $\alpha=\beta$ and $\alpha=\beta=0$, respectively, in the results given in §18.16(ii). For Legendre see also Hale and Townsend (2016, Lemma 2.3). For Chebyshev the zeros can be read from (18.5.1)–(18.5.4). See also (18.16.2), (18.16.3) or (3.5.23)–(3.5.25).

## §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 Ch.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 Ch.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 Ch.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 $(n+2)x_{n,1}\geq\left(n-1-\sqrt{n^{2}+(n+2)(\alpha+1)}\right)^{2}-1,$ ⓘ Symbols: $n$: nonnegative integer and $x$: real variable Referenced by: §18.16(iv), Erratum (V1.0.5) for Equations (18.16.12), (18.16.13), Erratum (V1.2.0) for Equations (18.16.12), (18.16.13) 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). See also: Annotations for §18.16(iv), §18.16(iv), §18.16 and Ch.18
 18.16.13 $(n+2)x_{n,n}\leq\left(n-1+\sqrt{n^{2}+(n+2)(\alpha+1)}\right)^{2}-1.$ ⓘ Symbols: $n$: nonnegative integer and $x$: real variable Referenced by: §18.16(iv), Erratum (V1.0.5) for Equations (18.16.12), (18.16.13), Erratum (V1.2.0) for Equations (18.16.12), (18.16.13) Permalink: http://dlmf.nist.gov/18.16.E13 Encodings: TeX, pMML, png Modification (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). See also: Annotations for §18.16(iv), §18.16(iv), §18.16 and Ch.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 $\operatorname{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 Ch.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 $\operatorname{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 Ch.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 $H_{n}\left(x\right)$.

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).

## §18.16(vii) Discriminants

The discriminant (18.2.20) can be given explicitly for classical OP’s.

### Jacobi

 18.16.19 $\operatorname{Disc}\left(P^{(\alpha,\beta)}_{n}\right)=2^{-n(n-1)}\prod_{j=1}^% {n}j^{j-2n+2}(j+\alpha)^{j-1}(j+\beta)^{j-1}(n+j+\alpha+\beta)^{n-j}.$ ⓘ Symbols: $\operatorname{Disc}\left(\NVar{x}\right)$: discriminant function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial and $n$: nonnegative integer Proved: Ismail (2009, (3.4.16))(proved) Referenced by: Erratum (V1.2.0) §18.16 Permalink: http://dlmf.nist.gov/18.16.E19 Encodings: TeX, pMML, png See also: Annotations for §18.16(vii), §18.16(vii), §18.16 and Ch.18

### Laguerre

 18.16.20 $\operatorname{Disc}\left(L^{(\alpha)}_{n}\right)=\prod_{j=1}^{n}j^{j-2n+2}(j+% \alpha)^{j-1}.$ ⓘ Symbols: $\operatorname{Disc}\left(\NVar{x}\right)$: discriminant function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial and $n$: nonnegative integer Proved: Ismail (2009, (3.4.15))(proved) Permalink: http://dlmf.nist.gov/18.16.E20 Encodings: TeX, pMML, png See also: Annotations for §18.16(vii), §18.16(vii), §18.16 and Ch.18

### Hermite

 18.16.21 $\operatorname{Disc}\left(H_{n}\right)=2^{\frac{3}{2}n(n-1)}\prod_{j=1}^{n}j^{j}.$ ⓘ Symbols: $\operatorname{Disc}\left(\NVar{x}\right)$: discriminant function, $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial and $n$: nonnegative integer Proved: Ismail (2009, (3.4.14))(proved) Referenced by: Erratum (V1.2.0) §18.16 Permalink: http://dlmf.nist.gov/18.16.E21 Encodings: TeX, pMML, png See also: Annotations for §18.16(vii), §18.16(vii), §18.16 and Ch.18