18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.15 Asymptotic Approximations 18.17 Integrals

§18.16 Zeros

ⓘ

Referenced by:: Erratum (V1.2.0) §18.16
Permalink:: http://dlmf.nist.gov/18.16
See also:: Annotations for Ch.18

§18.16(i) Distribution
§18.16(ii) Jacobi
§18.16(iii) Ultraspherical, Legendre and Chebyshev
§18.16(iv) Laguerre
§18.16(v) Hermite
§18.16(vi) Additional References
§18.16(vii) Discriminants

§18.16(i) Distribution

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Permalink:: http://dlmf.nist.gov/18.16.i
See also:: Annotations for §18.16 and Ch.18

See §18.2(vi).

§18.16(ii) Jacobi

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Keywords:: Jacobi polynomials, zeros
Notes:: See Szegő (1975, Theorems 6.21.1, 6.21.2, 6.21.3, 6.3.2). For (18.16.6), (18.16.7) see Gatteschi (1987). For (18.16.8) see Frenzen and Wong (1985b).
Referenced by:: §18.16(iv), §18.16(iii), Erratum (V1.0.12) for Subsections 18.15(i) and 18.16(ii), Erratum (V1.0.5) for References, Erratum (V1.0.7) for References
Permalink:: http://dlmf.nist.gov/18.16.ii
Correction (effective with 1.0.12):: The reference Frenzen and Wong (1985a) was replaced by Frenzen and Wong (1985b).
Suggested 2016-07-05 by Adri Olde Daalhuis
Addition (effective with 1.0.7):: The reference to Driver and Jordaan (2013) has been added at the end of this subsection.
Addition (effective with 1.0.5):: The reference to Dimitrov and Nikolov (2010) has been added at the end of this subsection.
See also:: Annotations for §18.16 and Ch.18

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

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Keywords:: Bessel functions, and zeros of Jacobi polynomials, classical orthogonal polynomials, inequalities, zeros
See also:: Annotations for §18.16(ii), §18.16 and Ch.18

18.16.2		$\theta_{n,m}^{(-\frac{1}{2},\frac{1}{2})}=\frac{(m-\tfrac{1}{2})\pi}{n+\tfrac{% 1}{2}}\leq\theta_{n,m}^{(\alpha,\beta)}\leq\frac{m\pi}{n+\tfrac{1}{2}}=\theta_% {n,m}^{(\frac{1}{2},-\frac{1}{2})},$
$\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\in$ : element of, $(\NVar{a},\NVar{b})$ : open interval, $m$ : nonnegative integer, $n$ : nonnegative integer and $\theta_{n,m}$ : zeros Referenced by: §18.16(iii), Erratum (V1.2.0) for Equations (18.16.2), (18.16.3) Permalink: http://dlmf.nist.gov/18.16.E2 Encodings: TeX, pMML, png See also: Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

18.16.3		$\theta_{n,m}^{(-\frac{1}{2},-\frac{1}{2})}=\frac{(m-\tfrac{1}{2})\pi}{n}\leq% \theta_{n,m}^{(\alpha,\alpha)}\leq\frac{m\pi}{n+1}=\theta_{n,m}^{(\frac{1}{2},% \frac{1}{2})},$
$\alpha\in[-\tfrac{1}{2},\tfrac{1}{2}]$ , $m=1,2,\dots,\left\lfloor\frac{1}{2}n\right\rfloor$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\in$ : element of, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $(\NVar{a},\NVar{b})$ : open interval, $m$ : nonnegative integer, $n$ : nonnegative integer and $\theta_{n,m}$ : zeros Referenced by: §18.16(iii), Erratum (V1.2.0) for Equations (18.16.2), (18.16.3) Permalink: http://dlmf.nist.gov/18.16.E3 Encodings: TeX, pMML, png See also: Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

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

18.16.4		${\frac{\left(m+\tfrac{1}{2}(\alpha+\beta-1)\right)\pi}{\rho}<\theta_{n,m}<% \frac{m\pi}{\rho}},$
$\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\in$ : element of, $m$ : nonnegative integer, $n$ : nonnegative integer, $\rho$ and $\theta_{n,m}$ : zeros Permalink: http://dlmf.nist.gov/18.16.E4 Encodings: TeX, pMML, png See also: Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

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

18.16.5		$\theta_{n,m}>\frac{\left(m+\tfrac{1}{2}\alpha-\tfrac{1}{4}\right){\pi}}{n+% \alpha+\tfrac{1}{2}},$
$\alpha=\beta$ , $\alpha\in(-\tfrac{1}{2},\tfrac{1}{2})$ , $m=1,2,\dots,\left\lfloor\tfrac{1}{2}n\right\rfloor$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $(\NVar{a},\NVar{b})$ : open interval, $m$ : nonnegative integer, $n$ : nonnegative integer and $\theta_{n,m}$ : zeros Permalink: http://dlmf.nist.gov/18.16.E5 Encodings: TeX, pMML, png See also: Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

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	$\displaystyle\theta_{n,m}$	$\displaystyle\geq\frac{j_{\alpha,m}}{\left(\rho^{2}+\tfrac{1}{4}-\tfrac{1}{2}(% \alpha^{2}+\beta^{2})-{\pi}^{-2}(1-4\alpha^{2})\right)^{\frac{1}{2}}},$
$\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]$ , $m=1,2,\dots,\left\lfloor\tfrac{1}{2}n\right\rfloor$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\in$ : element of, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $m$ : nonnegative integer, $n$ : nonnegative integer, $\rho$ and $\theta_{n,m}$ : zeros Referenced by: §18.16(ii) Permalink: http://dlmf.nist.gov/18.16.E7 Encodings: TeX, pMML, png See also: Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

Asymptotic Behavior

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Keywords:: Jacobi polynomials, asymptotic approximations, classical orthogonal polynomials, zeros
See also:: Annotations for §18.16(ii), §18.16 and Ch.18

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		$\theta_{n,m}=\phi_{m}+\left(\left(\alpha^{2}-\tfrac{1}{4}\right)\frac{1-\phi_{% m}\cot\phi_{m}}{2\phi_{m}}-\tfrac{1}{4}(\alpha^{2}-\beta^{2})\tan\left(\tfrac{% 1}{2}\phi_{m}\right)\right)\frac{1}{\rho^{2}}+\phi_{m}^{2}O\left(\frac{1}{\rho% ^{3}}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\cot\NVar{z}$ : cotangent function, $\tan\NVar{z}$ : tangent function, $m$ : nonnegative integer, $n$ : nonnegative integer, $\rho$ , $\theta_{n,m}$ : zeros and $\phi_{m}$ Referenced by: §18.16(ii) Permalink: http://dlmf.nist.gov/18.16.E8 Encodings: TeX, pMML, png See also: Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

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

Other Bounds

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See also:: Annotations for §18.16(ii), §18.16 and Ch.18

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

§18.16(iii) Ultraspherical, Legendre and Chebyshev

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Keywords:: Chebyshev polynomials, Legendre polynomials, ultraspherical polynomials, zeros
Referenced by:: Erratum (V1.2.0) §18.16
Permalink:: http://dlmf.nist.gov/18.16.iii
See also:: Annotations for §18.16 and Ch.18

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

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Keywords:: Laguerre polynomials, zeros
Notes:: For (18.16.10) and (18.16.11) see Szegő (1975, Theorem 6.31.3). For (18.16.12) and (18.16.13) see Dimitrov and Nikolov (2010). For (18.16.14) see Tricomi (1949). For (18.16.15) see Szegő (1975, Theorem 6.32). See also Gatteschi (2002).
Referenced by:: Erratum (V1.0.5) for References, Erratum (V1.0.7) for References, Erratum (V1.0.7) for Other
Permalink:: http://dlmf.nist.gov/18.16.iv
Correction (effective with 1.0.7):: Originally one of the conditions for the validity of (18.16.10), $n=1,2,\dots,m$ , was incorrect. The correct condition is $m=1,2,\dots,n$ . Also, the reference to Driver and Jordaan (2013) has been added after (18.16.13).
See also:: Annotations for §18.16 and Ch.18

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

18.16.9		$0<x_{n,1}<x_{n,2}<\cdots<x_{n,n}.$
ⓘ 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

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See also:: Annotations for §18.16(iv), §18.16 and Ch.18

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

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Keywords:: Laguerre polynomials, asymptotic behavior, classical orthogonal polynomials, inequalities, zeros
See also:: Annotations for §18.16(iv), §18.16 and Ch.18

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

18.16.14		$x_{n,n-m+1}=\nu+2^{\frac{2}{3}}a_{m}\nu^{\frac{1}{3}}+\tfrac{1}{5}2^{\frac{4}{% 3}}{a_{m}}^{2}\nu^{-\frac{1}{3}}+O\left(n^{-1}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $m$ : nonnegative integer, $n$ : nonnegative integer, $\nu$ and $x$ : real variable Referenced by: §18.16(iv) Permalink: http://dlmf.nist.gov/18.16.E14 Encodings: TeX, pMML, png See also: Annotations for §18.16(iv), §18.16(iv), §18.16 and Ch.18

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		$x_{n,m}<\nu+2^{\frac{2}{3}}a_{m}\nu^{\frac{1}{3}}+2^{-\frac{2}{3}}{a_{m}}^{2}% \nu^{-\frac{1}{3}},$
ⓘ Symbols: $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $m$ : nonnegative integer, $n$ : nonnegative integer, $\nu$ and $x$ : real variable Referenced by: §18.16(iv) Permalink: http://dlmf.nist.gov/18.16.E15 Encodings: TeX, pMML, png See also: Annotations for §18.16(iv), §18.16(iv), §18.16 and Ch.18

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

§18.16(v) Hermite

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Keywords:: Hermite polynomials, asymptotic approximations, asymptotic behavior, classical orthogonal polynomials, zeros
Notes:: See Szegő (1975, Theorem 6.32) and Sun (1996).
Permalink:: http://dlmf.nist.gov/18.16.v
See also:: Annotations for §18.16 and Ch.18

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		$x_{n,m}=(2n+1)^{\frac{1}{2}}+2^{-\frac{1}{3}}(2n+1)^{-\frac{1}{6}}a_{m}+% \epsilon_{n,m},$
ⓘ Symbols: $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $m$ : nonnegative integer, $n$ : nonnegative integer, $\epsilon_{n,m}$ and $x$ : real variable Permalink: http://dlmf.nist.gov/18.16.E17 Encodings: TeX, pMML, png See also: Annotations for §18.16(v), §18.16 and Ch.18

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 $H_{n}\left(x\right)$ .

§18.16(vi) Additional References

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Permalink:: http://dlmf.nist.gov/18.16.vi
See also:: Annotations for §18.16 and Ch.18

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

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Keywords:: classical orthogonal polynomials, discriminant
Referenced by:: Erratum (V1.2.0) §18.16, Version 1.2.0 (March 27, 2024)
Permalink:: http://dlmf.nist.gov/18.16.vii
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §18.16 and Ch.18

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

Jacobi

ⓘ

See also:: Annotations for §18.16(vii), §18.16 and Ch.18

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

ⓘ

See also:: Annotations for §18.16(vii), §18.16 and Ch.18

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

ⓘ

See also:: Annotations for §18.16(vii), §18.16 and Ch.18

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

18.15 Asymptotic Approximations 18.17 Integrals

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§18.16 Zeros

Contents

§18.16(i) Distribution

§18.16(ii) Jacobi

Inequalities

Asymptotic Behavior

Other Bounds

§18.16(iii) Ultraspherical, Legendre and Chebyshev

§18.16(iv) Laguerre

Inequalities

Asymptotic Behavior

§18.16(v) Hermite

§18.16(vi) Additional References

§18.16(vii) Discriminants

Jacobi

Laguerre

Hermite