Krawtchouk多项式的渐近性态

在前人的基础上,对Krawtchouk多项式及其零点的渐近性态进行了研究.首先推导出对于任意固定的u=n/N∈(0,P)或(0,q)Krawtchouk多项式Kn(λN)(其中λ=xN,0<λ<1)的一致有效渐近展开式.然后又得到了它的零点的渐近性态,并对其相应的误差限进行分析.该误差限为o(n-4/3).     

ASYMPTOTIC EXPANSIONS OF ZEROS FOR KRAWTCHOUK POLYNOMIALS WITH ERROR BOUNDS

Krawtchouk polynomials are frequently applied in modern physics. Based on the results which were educed by Li and Wong, the asymptotic expansions of Krawtchouk polynomials are improved by using Airy function, and uniform asymptotic expansions are got. Furthermore, the asymptotic expansions of the zeros for Krawtchouk polynomials are again deduced by using the property of the zeros of Airy function, and their corresponding error bounds are discussed. The obtained results give the asymptotic property of Krawtchouk polynomials with their zeros, which are better than the results educed by Li and Wong.     

关于Jacobi函数的渐近性态研究

采取改进取点x(t)的做法,提高了Jacobi函数的一项近似精确度.我们分别取x(t)的两项和三项,做出了Jacobi函数φ(α,β)μ(t) (α>-1 )当μ→+∞渐近近似,并给出了相应的误差限.随着x(t)取的项数增加,即点x(t)取的更“精确”,Jacobi函数φ(α,β)μ(t)渐近近似的精确度也随之提高.     

Krawtchouk多项式零点的渐近展开及其误差限

Krawtchouk多项式在现代物理学中有着广泛应用．基于Li和Wong的结果,利用Airy函数改进了Krawtchouk多项式的渐近展开式,而且得到了一个一致有效的渐近展开式A·D2进一步,利用Airy函数零点的性质,推导出了Krawtchouk多项式零点的渐近展开式,并讨论了其相应的误差限．同时还给出了Krawtchouk多项式和其零点的渐近性态,它优于Li和Wong的结果．