第二类Stirling数的一些同余式

Some Congruences for Stirling Numbers of the Second Kind

设k,n为非负整数,S(n,k)表示第二类Stirling数.本文研究了S(n,k)模2的方幂的同余式,首先给出了一类二项式系数模2的同余式,然后利用上述结果得到了S(n,a2~m+b)模2~m的同余式.其表达式均由简单二项式系数组成,其中m≥3,b=0,1,2.这些结果改进了Chan和Manna的结果.

Let A;and n be nonnegative integers.In this paper,we investigate the congruences for the Stirling numbers of the second kind S(n,k) modulo powers of 2.We first give a congruence for some certain binomial coefficients modulo 2.Using this result we then establish the congruences for S(n,a2~m + b) modulo 2~m,and express them in terms of binomial coefficients,where m ≥ 3,b = 0,1,2.These strengthen the results obtained by Chan and Manna.