关于组合幂和式的渐近展开  

Asymptotic expansion on combinatorial sums

作　　者：乌云高娃[1,2] 王天明[1] 

机构地区：[1]大连理工大学应用数学系,辽宁大连116024 [2]内蒙古大学数学科学学院,内蒙古呼和浩特010021

出　　处：《大连理工大学学报》2009年第3期463-468,共6页Journal of Dalian University of Technology

摘　　要：利用发生函数方法给出了组合和式∑nk=0nkkl的精确公式,从而得到了与第二类Stirling数有关的恒等式,并且进一步研究了组合幂和式∑nk=0nkrkl和∑nk=1nkr1kl的渐近性.首先借助于分析学工具和渐近估计方法定出整个和式起决定性作用的部分∑k∈Ankkl(其中A={k:1/2-ε≤k/n≤1/2+ε}),然后通过Euler-Maclaurin求和公式估计和式∑k∈Ankkl,最后给出了组合幂和式的完全渐近展开式.By means of the generating function method, the explicit formula of n∑k-0（nk）kl is given, so that an identity related to the Stirling numbers of the second kind is established. Moreover, the asymptotic expansions of the combinatorial sums n∑k=0（nk）krl and n∑k=1（nk）r1/k＇ are studied. To do these work, the determinant part∑k∈A（nk）kl of the whole sum is first ascertained, where A={k：1/2-ε≤k/n≤1/2＋ε}based on the theory of calculus and the asymptotic method. Next by using of the Euler-Maclaurin formula, the sum∑k∈A（nk）kl is estimated and finally the full asymptotic expansions of the combinatorial sums are obtained.

关 键 词：组合和式 STIRLING数 渐近展开 

分 类 号：O157.1[理学—数学]