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A uniform asymptotic formula for orthogonal polynomials associated with exp(-x 4 ). (English) Zbl 0952.41018

The paper is devoted to construct an asymptotic approximation for the orthogonal polynomials p n (x), associated with the Freud weight exp(-x 4 ), x. The following theorem extends a result of Nevai.

Theorem: Let 0<ε<1 and 0<M< be fixed, and let x=(4n/3) 1/4 w and Λ=4n/3. Then the asymptotic formula

p n (x)exp-x 4 2=2Λ 1/24 ζ w 2 -1 1/4 AiΛ 2/3 ζ+H(ζ) Λ 1/3 +O(n -ρ ),

holds uniformly for ρ=1, -1+εw=xλ -1 M, where

ζ(w)=-(9 8cos -1 w-3 8w(2w 2 +1)1-ω 2 ) 2/3 ,-1<w<1(8 8w(2w 2 +1)w 2 -1-9 8cosh -1 w) 2/3 ,w1
H(ζ)=-1 2ζ 1/2 cosh -1 w,ifw1;H(ζ)=-1 2(-ζ) 1/2 cos -1 w,if|w|<1,(1)

and Ai(x) is the Airy function. Moreover, when -1+εw1-ε, the uniform asymptotic formula (1) holds for ρ=7/6. Let b n =(4n/3). The following result is proved for the positive zeros x n,k of the polynomial p n (x)

x n,k =b n 1/4 +a ˜ k b n -5/12 18 -1/3 +b n -9/12 6 -1 -19a ˜ k 2 b n -13/12 90 -1 2 -2/3 3 1/3 +O(n -17/12 ),

where a ˜ k is the kth negative zero of the Airy function Ai(x).


MSC:
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
42C05General theory of orthogonal functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type