Krawtchouk多项式零点的渐近展开及其误差限

基金项目: 北京市中青年骨干教师培养计划资助项目;北京市教员委员会科技发展计划面上资助项目(KM200310015060)

Asymptotic Expansions of Zeros for Krawtchouk Polynomials With Error Bounds

Department of Basic Science, Beijing Institute of Graphic Communication, Beijing 102600, P. R. China

摘要: 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．
- Krawtchouk多项式 /
- 渐近展开 /
- 零点 /
- 误差限
Abstract: 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 were improved by using Airy function, and the uniform asymptotic expansions were got. Furthermore, the asymptotic expansions of the zeros for Krawt chouk polynomials were again deduced by using the property of the zeros of Airy function, and their corresponding error bounds were 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.
- Krawtchouk polynomial /
- asymptotic expansion /
- zero /
- error bounds

[1]	Sharapudinov I I. Asymptotic properties of Krawtchouk polynomials[J].Mat Zametki，1988,44(5):682—693;English translation,Math Notes,1988,44(5):855—862.
[2]	Ismail M E H, Simeonov P. Strong asymptotics for Krawtchouk polynomials[J].J Comput Appl Math，1998,100(2):121—144. doi: 10.1016/S0377-0427(98)00183-6
[3]	Li X C, Wong R. A uniform asymptotic expansion for Krawtchouk polynomials[J].Journal of Approximation Theory,2000,106(1):155—184. doi: 10.1006/jath.2000.3474
[4]	Olver F W J.Uniform asymptotic expansions for Weber parabolic cylinder functions of large order[J].J Res Nat Bur Standards Sect B,1959,63(2):131—169.
[5]	Hethcote H W. Error bounds for asymptotic approximations of zeros of transcendental functions[J].SIAM J Math Anal,1970,1(2):147—152. doi: 10.1137/0501015
[6]	Ma E J,Zhu X F，Li X C. Asymptotic approximations of Jacobi polynomials and their zeros with error bounds[J].Tsinghua Science and Technology,2003,8(1):71—78.
[7]	朱晓峰，李秀淳.关于Jacobi函数渐近性态[J]. 数学的实践与认识，2005，35(4):200—205.
[8]	Zhu X F, Li X C.Asymptotic approximations of Jacobi polynomials and their zeros with error bounds[J].Acta Mathematica Sinica,2006,22(3):729—740. doi: 10.1007/s10114-005-0593-6