18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.15 Asymptotic Approximations 18.17 Integrals

§18.16 Zeros

Permalink:: http://dlmf.nist.gov/18.16

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

§18.16(i) Distribution

Permalink:: http://dlmf.nist.gov/18.16.i

See §18.2(vi).

§18.16(ii) Jacobi

Notes:: See Szegö (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 (1985).
Keywords:: Jacobi polynomials
Referenced by:: ¶ ‣ §18.16(iv), §18.16(iii), Other Changes
Permalink:: http://dlmf.nist.gov/18.16.ii
Addition (effective with 1.0.5):: The reference to Dimitrov and Nikolov (2010) has been added at the end of this subsection.
Reported 2012-07-30

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:: : nonnegative integer and $\theta_{{n,m}}$ : zeros
Permalink:: http://dlmf.nist.gov/18.16.E1
Encodings:: TeX, pMML, png

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

Keywords:: classical orthogonal polynomials, inequalities

18.16.2 $\frac{(m-\tfrac{1}{2})\pi}{n+\tfrac{1}{2}}\leq\theta_{{n,m}}\leq\frac{m\pi}{n+% \tfrac{1}{2}},$ $\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]$ ,

Symbols:: : closed interval, $\in$ : element of, : nonnegative integer, : nonnegative integer and $\theta_{{n,m}}$ : zeros
Permalink:: http://dlmf.nist.gov/18.16.E2
Encodings:: TeX, pMML, png

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

Symbols:: : closed interval, $\in$ : element of, $\left\lfloor x\right\rfloor$ : floor of , : nonnegative integer, : nonnegative integer and $\theta_{{n,m}}$ : zeros
Permalink:: http://dlmf.nist.gov/18.16.E3
Encodings:: TeX, pMML, png

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:: : closed interval, $\in$ : element of, : nonnegative integer, : nonnegative integer, $\rho$ and $\theta_{{n,m}}$ : zeros
Permalink:: http://dlmf.nist.gov/18.16.E4
Encodings:: TeX, pMML, png

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:: $\in$ : element of, $\left\lfloor x\right\rfloor$ : floor of , : open interval, : nonnegative integer, : nonnegative integer and $\theta_{{n,m}}$ : zeros
Permalink:: http://dlmf.nist.gov/18.16.E5
Encodings:: TeX, pMML, png

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

18.16.6 $\theta_{{n,m}}\leq\frac{j_{{\alpha,m}}}{\left(\rho^{2}+\tfrac{1}{12}\left(1-% \alpha^{2}-3\beta^{2}\right)\right)^{{\frac{1}{2}}}},$ $\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]$ ,

Symbols:: : closed interval, $\in$ : element of, : nonnegative integer, : 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

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

Symbols:: : closed interval, $\in$ : element of, $\left\lfloor x\right\rfloor$ : floor of , : nonnegative integer, : 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

¶ Asymptotic Behavior

Keywords:: Jacobi polynomials, classical orthogonal polynomials

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{\tan\/}\nolimits z$ : tangent function, : nonnegative integer, : 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

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

¶ Other Bounds

See Dimitrov and Nikolov (2010).

§18.16(iii) Ultraspherical and Legendre

Keywords:: Legendre polynomials, ultraspherical polynomials
Permalink:: http://dlmf.nist.gov/18.16.iii

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

Notes:: For (18.16.10) and (18.16.11) see Szegö (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 Szegö (1975, Theorem 6.32). See also Gatteschi (2002).
Keywords:: Laguerre polynomials
Referenced by:: Other Changes
Permalink:: http://dlmf.nist.gov/18.16.iv

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

Symbols:: : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.16.E9
Encodings:: TeX, pMML, png

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

¶ Inequalities

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

18.16.10 $x_{{n,m}}>\ifrac{j_{{\alpha,m}}^{2}}{\nu},$

Symbols:: : nonnegative integer, : nonnegative integer, $\nu$ and : real variable
Referenced by:: ¶ ‣ §18.16(iv), §18.16(iv)
Permalink:: http://dlmf.nist.gov/18.16.E10
Encodings:: TeX, pMML, png

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:: : nonnegative integer, : nonnegative integer, $\nu$ and : real variable
Referenced by:: §18.16(iv)
Permalink:: http://dlmf.nist.gov/18.16.E11
Encodings:: TeX, pMML, png

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:: : nonnegative integer and : 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

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:: : nonnegative integer and : 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

¶ Asymptotic Behavior

Keywords:: Laguerre polynomials, classical orthogonal polynomials, inequalities

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{a_{{k}}\/}\nolimits$ : ^th zero of Airy $\mathop{\mathrm{Ai}\/}\nolimits$ , : nonnegative integer, : nonnegative integer, $\nu$ and : real variable
Referenced by:: §18.16(iv)
Permalink:: http://dlmf.nist.gov/18.16.E14
Encodings:: TeX, pMML, png

where $\mathop{a_{{m}}\/}\nolimits$ is the 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 $x_{{n,m}}<\nu+2^{{\frac{2}{3}}}\mathop{a_{{m}}\/}\nolimits\nu^{{\frac{1}{3}}}+% 2^{{-\frac{2}{3}}}{\mathop{a_{{m}}\/}\nolimits^{{2}}}\nu^{{-\frac{1}{3}}},$

Symbols:: $\mathop{a_{{k}}\/}\nolimits$ : ^th zero of Airy $\mathop{\mathrm{Ai}\/}\nolimits$ , : nonnegative integer, : nonnegative integer, $\nu$ and : real variable
Referenced by:: §18.16(iv)
Permalink:: http://dlmf.nist.gov/18.16.E15
Encodings:: TeX, pMML, png

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

§18.16(v) Hermite

Notes:: See Szegö (1975, Theorem 6.32) and Sun (1996).
Keywords:: Hermite polynomials, classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.16.v

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:: $\left\lfloor x\right\rfloor$ : floor of , : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.16.E16
Encodings:: TeX, pMML, png

Then

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

Symbols:: $\mathop{a_{{k}}\/}\nolimits$ : ^th zero of Airy $\mathop{\mathrm{Ai}\/}\nolimits$ , : nonnegative integer, : nonnegative integer, $\epsilon_{{n,m}}$ and : real variable
Permalink:: http://dlmf.nist.gov/18.16.E17
Encodings:: TeX, pMML, png

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

18.16.18 $\epsilon_{{n,m}}=\mathop{O\/}\nolimits\!\left(n^{{-\frac{5}{6}}}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, : nonnegative integer, : nonnegative integer and $\epsilon_{{n,m}}$
Permalink:: http://dlmf.nist.gov/18.16.E18
Encodings:: TeX, pMML, png

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 $\alpha=\mu^{{-\frac{4}{3}}}\mathop{a_{{m}}\/}\nolimits$ . 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 $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ .

§18.16(vi) Additional References

Permalink:: http://dlmf.nist.gov/18.16.vi

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 18.15 Asymptotic Approximations 18.17 Integrals

§18.16 Zeros

Contents

§18.16(i) Distribution

§18.16(ii) Jacobi

¶ Inequalities

¶ Asymptotic Behavior

¶ Other Bounds

§18.16(iii) Ultraspherical and Legendre

§18.16(iv) Laguerre

¶ Inequalities

¶ Asymptotic Behavior

§18.16(v) Hermite

§18.16(vi) Additional References