Digital Library of Mathematical Functions
About the Project
NIST
18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.16 Zeros

Contents

§18.16(i) Distribution

See §18.2(vi).

§18.16(ii) Jacobi

Let θn,m, m=1,2,,n, denote the zeros of Pn(α,β)(cosθ) with

18.16.1 0<θn,1<θn,2<<θn,n<π.

Then θn,m is strictly increasing in α and strictly decreasing in β; furthermore, if α=β, then θn,m is strictly increasing in α.

Inequalities

18.16.2 (m-12)πn+12θn,mmπn+12,
α,β[-12,12],
18.16.3 (m-12)πnθn,mmπn+1,
α=β, α[-12,12], m=1,2,,12n.

Also, with ρ defined as in (18.15.5)

18.16.4 (m+12(α+β-1))πρ<θn,m<mπρ,
α,β[-12,12],

except when α2=β2=14.

18.16.5 θn,m>(m+12α-14)πn+α+12,
α=β, α(-12,12), m=1,2,,12n.

Let jα,m be the mth positive zero of the Bessel function Jα(x)10.21(i)). Then

18.16.6 θn,m jα,m(ρ2+112(1-α2-3β2))12,
α,β[-12,12],
18.16.7 θn,m jα,m(ρ2+14-12(α2+β2)-π-2(1-4α2))12,
α,β[-12,12], m=1,2,,12n.

Asymptotic Behavior

Let ϕm=jα,m/ρ. Then as n, with α (>-12) and β (-1-α) fixed,

18.16.8 θn,m=ϕm+((α2-14)1-ϕmcotϕm2ϕm-14(α2-β2)tan(12ϕm))1ρ2+ϕm2O(1ρ3),

uniformly for m=1,2,,cn, where c is an arbitrary constant such that 0<c<1.

Other Bounds

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

§18.16(iii) Ultraspherical and Legendre

For ultraspherical and Legendre polynomials, set α=β and α=β=0, respectively, in the results given in §18.16(ii).

§18.16(iv) Laguerre

The zeros of Ln(α)(x) are denoted by xn,m, m=1,2,,n, with

18.16.9 0<xn,1<xn,2<<xn,n.

Also, ν is again defined by (18.15.17).

Inequalities

For m=1,2,,n, and with jα,m as in §18.16(ii),

18.16.10 xn,m>jα,m2/ν,
18.16.11 xn,m<(4m+2α+2)(2m+α+1+((2m+α+1)2+14-α2)12)/ν.

The constant jα,m2 in (18.16.10) is the best possible since the ratio of the two sides of this inequality tends to 1 as n.

For the smallest and largest zeros we have

18.16.12 xn,12n2+αn-n+2α+2-2(n-1)n2+(n+2)(α+1)n+2,
18.16.13 xn,n2n2+αn-n+2α+2+2(n-1)n2+(n+2)(α+1)n+2.

See Driver and Jordaan (2013).

Asymptotic Behavior

As n, with α and m fixed,

18.16.14 xn,n-m+1=ν+223amν13+15243am2ν-13+O(n-1),

where am is the mth negative zero of Ai(x)9.9(i)). For three additional terms in this expansion see Gatteschi (2002). Also,

18.16.15 xn,m<ν+223amν13+2-23am2ν-13,

when α(-12,12).

§18.16(v) Hermite

All zeros of Hn(x) lie in the open interval (-2n+1,2n+1). In view of the reflection formula, given in Table 18.6.1, we may consider just the positive zeros xn,m, m=1,2,,12n. Arrange them in decreasing order:

18.16.16 (2n+1)12>xn,1>xn,2>>xn,n/2>0.

Then

18.16.17 xn,m=(2n+1)12+2-13(2n+1)-16am+ϵn,m,

where am is the mth negative zero of Ai(x)9.9(i)), ϵn,m<0, and as n with m fixed

18.16.18 ϵn,m=O(n-56).

For an asymptotic expansion of xn,m as n that applies uniformly for m=1,2,,12n, see Olver (1959, §14(i)). In the notation of this reference xn,m=ua,m, μ=2n+1, and α=μ-43am. 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 Ln(±12)(x) lead immediately to results for the zeros of Hn(x).

§18.16(vi) Additional References

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).