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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.14 Inequalities

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§18.14(i) Upper Bounds

Jacobi

18.14.1 |Pn(α,β)(x)|Pn(α,β)(1)=(α+1)nn!,
-1x1, αβ>-1, α-12,
18.14.2 |Pn(α,β)(x)||Pn(α,β)(-1)|=(β+1)nn!,
-1x1, βα>-1, β-12.
18.14.3 (12(1-x))12α+14(12(1+x))12β+14|Pn(α,β)(x)|Γ(max(α,β)+n+1)π12n!(n+12(α+β+1))max(α,β)+12,
-1x1, -12α12, -12β12.

Ultraspherical

18.14.4 |Cn(λ)(x)|Cn(λ)(1)=(2λ)nn!,
-1x1, λ>0.
18.14.5 |C2m(λ)(x)||C2m(λ)(0)|=|(λ)mm!|,
-1x1, -12<λ<0,
18.14.6 |C2m+1(λ)(x)|<-2(λ)m+1((2m+1)(2λ+2m+1))12m!,
-1x1, -12<λ<0.
18.14.7 (n+λ)1-λ(1-x2)12λ|Cn(λ)(x)|<21-λΓ(λ),
-1x1, 0<λ<1.

Laguerre

18.14.8 e-12x|Ln(α)(x)|Ln(α)(0)=(α+1)nn!,
0x<, α0.

Hermite

18.14.9 1(2nn!)12e-12x2|Hn(x)|1,
-<x<.

For further inequalities see Abramowitz and Stegun (1964, §22.14).

§18.14(ii) Turan-Type Inequalities

Legendre

18.14.10 (Pn(x))2Pn-1(x)Pn+1(x),
-1x1.

Jacobi

Let Rn(x)=Pn(α,β)(x)/Pn(α,β)(1). Then

18.14.11 (Rn(x))2Rn-1(x)Rn+1(x),
-1x1, βα>-1.

Laguerre

18.14.12 (Ln(α)(x))2Ln-1(α)(x)Ln+1(α)(x),
0x<, α0.

Hermite

18.14.13 (Hn(x))2Hn-1(x)Hn+1(x),
-<x<.

§18.14(iii) Local Maxima and Minima

Jacobi

Let the maxima xn,m, m=0,1,,n, of |Pn(α,β)(x)| in [-1,1] be arranged so that

18.14.14 -1=xn,0<xn,1<<xn,n-1<xn,n=1.

When (α+12)(β+12)>0 choose m so that

18.14.15 xn,m(β-α)/(α+β+1)xn,m+1.

Then

18.14.16 |Pn(α,β)(xn,0)| >|Pn(α,β)(xn,1)|
>
>|Pn(α,β)(xn,m)|,
|Pn(α,β)(xn,n)| >|Pn(α,β)(xn,n-1)|
>
>|Pn(α,β)(xn,m+1)|,
α>-12,β>-12.
18.14.17 |Pn(α,β)(xn,0)| <|Pn(α,β)(xn,1)|
<
<|Pn(α,β)(xn,m)|,
|Pn(α,β)(xn,n)| <|Pn(α,β)(xn,n-1)|
<
<|Pn(α,β)(xn,m+1)|,
-1<α<-12,-1<β<-12.

Also,

18.14.18 |Pn(α,β)(xn,0)|<|Pn(α,β)(xn,1)|<<|Pn(α,β)(xn,n)|,
α-12, -1<β-12,
18.14.19 |Pn(α,β)(xn,0)|>|Pn(α,β)(xn,1)|>>|Pn(α,β)(xn,n)|,
β-12, -1<α-12,

except that when α=β=-12 (Chebyshev case) |Pn(α,β)(xn,m)| is constant.

Szegö–Szász Inequality

18.14.20 |Pn(α,β)(xn,n-m)Pn(α,β)(1)|>|Pn+1(α,β)(xn+1,n-m+1)Pn+1(α,β)(1)|,
α=β>-12, m=1,2,,n.

For extensions of (18.14.20) see Askey (1990) and Wong and Zhang (1994a, b).

Laguerre

Let the maxima xn,m, m=0,1,,n-1, of |Ln(α)(x)| in [0,) be arranged so that

18.14.21 0=xn,0<xn,1<<xn,n-1<xn,n=.

When α>-12 choose m so that

18.14.22 xn,mα+12xn,m+1.

Then

18.14.23 |Ln(α)(xn,0)| >|Ln(α)(xn,1)|
>
>|Ln(α)(xn,m)|,
|Ln(α)(xn,n-1)| >|Ln(α)(xn,n-2)|
>
>|Ln(α)(xn,m+1)|.

Also, when α-12

18.14.24 |Ln(α)(xn,0)|<|Ln(α)(xn,1)|<<|Ln(α)(xn,n-1)|.

Hermite

The successive maxima of |Hn(x)| form a decreasing sequence for x0, and an increasing sequence for x0.