§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)
Permalink:: http://dlmf.nist.gov/18.16.ii

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:: $n$ : nonnegative integer and $\theta _{{n,m}}$ : zeros
Permalink:: http://dlmf.nist.gov/18.16.E1
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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:: $[a,b]$ : closed interval, $\in$ : element of, $m$ : nonnegative integer, $n$ : nonnegative integer and $\theta _{{n,m}}$ : zeros
Permalink:: http://dlmf.nist.gov/18.16.E2
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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:: $[a,b]$ : closed interval, $\in$ : element of, $\left\lfloor x\right\rfloor$ : floor of $x$ , $m$ : nonnegative integer, $n$ : nonnegative integer and $\theta _{{n,m}}$ : zeros
Permalink:: http://dlmf.nist.gov/18.16.E3
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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:: $[a,b]$ : 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
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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 $x$ , $(a,b)$ : open interval, $m$ : nonnegative integer, $n$ : nonnegative integer and $\theta _{{n,m}}$ : zeros
Permalink:: http://dlmf.nist.gov/18.16.E5
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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 $\theta _{{n,m}}\leq\frac{j_{{\alpha,m}}}{\left(\rho^{2}+\tfrac{1}{12}\left(1-\alpha^{2}-3\beta^{2}\right)\right)^{{\frac{1}{2}}}},$ $\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]$ ,

Symbols:: $[a,b]$ : 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

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:: $[a,b]$ : closed interval, $\in$ : element of, $\left\lfloor x\right\rfloor$ : 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

¶ 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, $m$ : 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

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

§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 Ismail and Li (1992). For (18.16.14) see Tricomi (1949). For (18.16.15) see Szegö (1975, Theorem 6.32). See also Gatteschi (2002).
Keywords:: Laguerre polynomials
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:: $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.16.E9
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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:: $m$ : 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
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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
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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 $x_{{n,1}}>2n+\alpha-2-(1+4(n-1)(n+\alpha-1))^{{\frac{1}{2}}},$

Symbols:: $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.16(iv)
Permalink:: http://dlmf.nist.gov/18.16.E12
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18.16.13 $x_{{n,n}}<2n+\alpha-2+(1+4(n-1)(n+\alpha-1))^{{\frac{1}{2}}}.$

Symbols:: $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.16(iv)
Permalink:: http://dlmf.nist.gov/18.16.E13
Encodings:: TeX, pMML, png

¶ Asymptotic Behavior

Keywords:: Laguerre polynomials, classical orthogonal polynomials, inequalities

As $n\to\infty$ , with $\alpha$ and $m$ 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, $\mathop{a_{{k}}\/}\nolimits$ : $k$ th zero of Airy $\mathop{\mathrm{Ai}\/}\nolimits$ , $m$ : 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

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

Symbols:: $\mathop{a_{{k}}\/}\nolimits$ : $k$ th zero of Airy $\mathop{\mathrm{Ai}\/}\nolimits$ , $m$ : nonnegative integer, $n$ : nonnegative integer, $\nu$ and $x$ : real variable
Referenced by:: §18.16(iv)
Permalink:: http://dlmf.nist.gov/18.16.E15
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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 $x$ , $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.16.E16
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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:: $\mathop{a_{{k}}\/}\nolimits$ : $k$ th zero of Airy $\mathop{\mathrm{Ai}\/}\nolimits$ , $m$ : nonnegative integer, $n$ : nonnegative integer, $\epsilon _{{n,m}}$ and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.16.E17
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where $\mathop{a_{{m}}\/}\nolimits$ 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 $\epsilon _{{n,m}}=\mathop{O\/}\nolimits\!\left(n^{{-\frac{5}{6}}}\right).$

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

§18.16 Zeros

Contents

§18.16(i) Distribution

§18.16(ii) Jacobi

¶ Inequalities

¶ Asymptotic Behavior

§18.16(iii) Ultraspherical and Legendre

§18.16(iv) Laguerre

¶ Inequalities

¶ Asymptotic Behavior

§18.16(v) Hermite

§18.16(vi) Additional References