Krawtchouk多项式的渐近性态

ASYMPTOTIC PROPERTIES OF KRAWTCHOUK POLYNOMIALS

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v=n/N∈(0,p)或(0,q)Krawtchouk多项式K_n(λN)(其中λ=x/N,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).

In this paper, the asymptotic properties of Krawtchouk polynomials and their zeros are investigated. Firstly, a uniform asymptotic expansion for Krawtchouk polynomials Kn （λN） （λ =N/n,0 〈 λ 〈 1） is derived, for any fixed v = N/n∈ （0,q） or （0,p）. Then asymptotic expansions of their zeros are given. Finally the error bound O（n^-/34） is obtained.