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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.15 Asymptotic Approximations


§18.15(i) Jacobi

With the exception of the penultimate paragraph, we assume throughout this subsection that α, β, and M (=0,1,2,) are all fixed.

18.15.1 (sin12θ)α+12(cos12θ)β+12Pn(α,β)(cosθ)=π-122n+α+β+1B(n+α+1,n+β+1)(m=0M-1fm(θ)2m(2n+α+β+2)m+O(n-M)),

as n, uniformly with respect to θ[δ,π-δ]. Here, and elsewhere in §18.15, δ is an arbitrary small positive constant. Also, B(a,b) is the beta function (§5.12) and

18.15.2 fm(θ)==0mCm,(α,β)!(m-)!cosθn,m,(sin12θ)(cos12θ)m-,


18.15.3 Cm,(α,β)=(12+α)(12-α)(12+β)m-(12-β)m-,


18.15.4 θn,m,=12(2n+α+β+m+1)θ-12(α++12)π.

When α,β(-12,12), the error term in (18.15.1) is less than twice the first neglected term in absolute value. See Hahn (1980), where corresponding results are given when x is replaced by a complex variable z that is bounded away from the orthogonality interval [-1,1].

Next, let

18.15.5 ρ=n+12(α+β+1).

Then as n,

18.15.6 (sin12θ)α+12(cos12θ)β+12Pn(α,β)(cosθ)=Γ(n+α+1)212ραn!(θ12Jα(ρθ)m=0MAm(θ)ρ2m+θ32Jα+1(ρθ)m=0M-1Bm(θ)ρ2m+1+εM(ρ,θ)),

where Jν(z) is the Bessel function (§10.2(ii)), and

18.15.7 εM(ρ,θ)={θO(ρ-2M-(3/2)),cρ-1θπ-δ,θα+(5/2)O(ρ-2M+α),0θcρ-1,

with c denoting an arbitrary positive constant. Also,

18.15.8 A0(θ) =1,
θB0(θ) =14g(θ),
A1(θ) =18g(θ)-1+2α8g(θ)θ-132(g(θ))2,


18.15.9 g(θ)=(14-α2)(cot(12θ)-(12θ)-1)-(14-β2)tan(12θ).

For higher coefficients see Baratella and Gatteschi (1988), and for another estimate of the error term see Wong and Zhao (2003).

For large β, fixed α, and 0n/βc, Dunster (1999) gives asymptotic expansions of Pn(α,β)(z) that are uniform in unbounded complex z-domains containing z=±1. These expansions are in terms of Whittaker functions (§13.14). This reference also supplies asymptotic expansions of Pn(α,β)(z) for large n, fixed α, and 0β/nc. The latter expansions are in terms of Bessel functions, and are uniform in complex z-domains not containing neighborhoods of 1. For a complementary result, see Wong and Zhao (2004). By using the symmetry property given in the second row of Table 18.6.1, the roles of α and β can be interchanged.

For an asymptotic expansion of Pn(α,β)(z) as n that holds uniformly for complex z bounded away from [-1,1], see Elliott (1971). The first term of this expansion also appears in Szegő (1975, Theorem 8.21.7).

§18.15(ii) Ultraspherical

For fixed λ(0,1) and fixed M=0,1,2,,

18.15.10 Cn(λ)(cosθ)=22λΓ(λ+12)π12Γ(λ+1)(2λ)n(λ+1)n(m=0M-1(λ)m(1-λ)mm!(n+λ+1)mcosθn,m(2sinθ)m+λ+O(1nM)),

as n uniformly with respect to θ[δ,π-δ], where

18.15.11 θn,m=(n+m+λ)θ-12(m+λ)π.

For a bound on the error term in (18.15.10) see Szegő (1975, Theorem 8.21.11).

Asymptotic expansions for Cn(λ)(cosθ) can be obtained from the results given in §18.15(i) by setting α=β=λ-12 and referring to (18.7.1). See also Szegő (1933) and Szegő (1975, Eq. (8.21.14)).

§18.15(iii) Legendre

For fixed M=0,1,2,,

18.15.12 Pn(cosθ)=(2sinθ)12m=0M-1(-12m)(m-12n)cosαn,m(2sinθ)m+O(1nM+12),

as n, uniformly with respect to θ[δ,π-δ], where

18.15.13 αn,m=(n-m+12)θ+(n-12m-14)π.

Also, when 16π<θ<56π, the right-hand side of (18.15.12) with M= converges; paradoxically, however, the sum is 2Pn(cosθ) and not Pn(cosθ) as stated erroneously in Szegő (1975, §8.4(3)).

For these results and further information see Olver (1997b, pp. 311–313). Another expansion follows from (18.15.10) by taking λ=12; see Szegő (1975, Theorem 8.21.5).

For asymptotic expansions of Pn(cosθ) and Pn(coshξ) that are uniformly valid when 0θπ-δ and 0ξ< see §14.15(iii) with μ=0 and ν=n. These expansions are in terms of Bessel functions and modified Bessel functions, respectively.

§18.15(iv) Laguerre

In Terms of Elementary Functions

For fixed M=0,1,2,, and fixed α,

18.15.14 Ln(α)(x)=n12α-14e12xπ12x12α+14(cosθn(α)(x)(m=0M-1am(x)n12m+O(1n12M))+sinθn(α)(x)(m=1M-1bm(x)n12m+O(1n12M))),

as n, uniformly on compact x-intervals in (0,), where

18.15.15 θn(α)(x)=2(nx)12-(12α+14)π.

The leading coefficients are given by

18.15.16 a0(x) =1,
a1(x) =0,
b1(x) =148x12(4x2-12α2-24αx-24x+3).

See also Deaño et al. (2013).

In Terms of Bessel Functions


18.15.17 ν=4n+2α+2,
18.15.18 ξ=12(x-x2+arcsin(x)),

Then for fixed M=0,1,2,, and fixed α,

18.15.19 Ln(α)(νx)=e12νx2αx12α+14(1-x)14(ξ12Jα(νξ)m=0M-1Am(ξ)ν2m+ξ-12Jα+1(νξ)m=0M-1Bm(ξ)ν2m+1+ξ12envJα(νξ)O(1ν2M-1)),

as n uniformly for 0x1-δ. Here Jν(z) denotes the Bessel function (§10.2(ii)), envJν(z) denotes its envelope (§2.8(iv)), and δ is again an arbitrary small positive constant. The leading coefficients are given by A0(ξ)=1 and

18.15.20 B0(ξ)=-12(1-4α28+ξ(1-xx)12(4α2-18+14x1-x+524(x1-x)2)).

In Terms of Airy Functions

Again define ν as in (18.15.17); also,

18.15.21 ζ =-(34(arccos(x)-x-x2))23,
ζ =(34(x2-x-arccosh(x)))23,

Then for fixed M=0,1,2,, and fixed α,

18.15.22 Ln(α)(νx)=(-1)ne12νx2α-12x12α+14(ζx-1)14×(Ai(ν23ζ)ν13m=0M-1Em(ζ)ν2m+Ai(ν23ζ)ν53m=0M-1Fm(ζ)ν2m+envAi(ν23ζ)O(1ν2M-23)),

as n uniformly for δx<. Here Ai denotes the Airy function (§9.2), Ai denotes its derivative, and envAi denotes its envelope (§2.8(iii)). The leading coefficients are given by E0(ζ)=1 and

18.15.23 F0(ζ)=-548ζ2+(x-1xζ)12(12α2-18-14xx-1+524(xx-1)2),

§18.15(v) Hermite


18.15.24 μ=2n+1,
18.15.25 λn={Γ(n+1)/Γ(12n+1),n even,Γ(n+2)/(μ12Γ(12n+32)),n odd,


18.15.26 ωn,m(x)=μ12x-12(m+n)π.

Then for fixed M=0,1,2,,

18.15.27 Hn(x)=λne12x2(m=0M-1um(x)cosωn,m(x)μ12m+O(1μ12M)),

as n, uniformly on compact x-intervals on . The coefficients um(x) are polynomials in x, and u0(x)=1, u1(x)=16x3.

For more powerful asymptotic expansions as n in terms of elementary functions that apply uniformly when 1+δt<, -1+δt1-δ, or -<t-1-δ, where t=x/2n+1 and δ is again an arbitrary small positive constant, see §§12.10(i)12.10(iv) and 12.10(vi). And for asymptotic expansions as n in terms of Airy functions that apply uniformly when -1+δt< or -<t1-δ, see §§12.10(vii) and 12.10(viii). With μ=2n+1 the expansions in Chapter 12 are for the parabolic cylinder function U(-12μ2,μt2), which is related to the Hermite polynomials via

18.15.28 Hn(x)=214(μ2-1)e12μ2t2U(-12μ2,μt2);

compare (18.11.3).

For an error bound for the first term in the Airy-function expansions see Olver (1997b, p. 403).

See also Geronimo et al. (2004).

§18.15(vi) Other Approximations

The asymptotic behavior of the classical OP’s as x± with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1.

For asymptotic approximations of Jacobi, ultraspherical, and Laguerre polynomials in terms of Hermite polynomials, see López and Temme (1999a). These approximations apply when the parameters are large, namely α and β (subject to restrictions) in the case of Jacobi polynomials, λ in the case of ultraspherical polynomials, and |α|+|x| in the case of Laguerre polynomials. See also Dunster (1999), Atia et al. (2014) and Temme (2015, Chapter 32).