渐近于Hermite多项式的双正交系统

本文利用渐近于Gauss函数的函数类?,给出渐近于Hermite正交多项式的一类Appell多项式的构造方法,使得该序列与?的n阶导数之间构成了一组双正交系统.利用此结果,本文得到多种正交多项式和组合多项式的渐近性质.特别地,由N阶B样条所生成的Appell多项式序列恰为N阶Bernoulli多项式.从而,Bernoulli多项式与B样条的导函数之间构成了一组双正交系统,且标准化之后的Bernoulli多项式的渐近形式为Hermite多项式.由二项分布所生成的Appell序列为Euler多项式,从而,Euler多项式与二项分布的导函数之间构成一组双正交系统,且标准化之后的Euler多项式渐近于Hermite多项式.本文给出Appell序列的生成函数满足的尺度方程的充要条件,给出渐近于Hermite多项式的函数列的判定定理.应用该定理,验证广义Buchholz多项式、广义Laguerre多项式和广义Ultraspherical(Gegenbauer)多项式渐近于Hermite多项式的性质,从而验证超几何多项式的Askey格式的成立.

Approximation of Hermite polynomials by biorthognal systerms

In this paper, the structure to a family of Appell sequences that approximate to Hermite polynomials is investigated by the functional Ф which approximates to Gaussian function to construct the biothogonal systems between the sequences and the derivatives of Ф Therefore, the asymptotic relations between several orthogonal polynomials and combinatoric polynomials are derived from the biothogonal systems. Especially, the Appell sequences generated by the uniform B-splines of order N are Bernoulli polynomials of order N which indicate the biorthogonal relationship between Bernoulli polynomials and the derivatives of B-splines. Therefore, the standardized Bernoulli polynomials approximate to Hermite polynomials. The asymptotic properties of standardized Euler polynomials to Hermite polynomials are derived by the biothognal systems generated by the binomial distribution and Euler polynomials. The judging theorem of the approximation to Hermite polynomials by a sequence of functions and the necessary and sufficient condition of the generating functions to the Appell sequence which satisfies the scaling equations are also discussed. The asymptotic representations of generalized Buchholz, Laguerre and Ultraspherical （Gegenbauer） polynomials to Hermite polynomials are proved by the theorems which in turn verify the Askey scheme of hypergeometric orthogonal polynomials.