关于多项式系数的整除性

On the Divisibility of Multinomial Number

设 n是正整数 ,k1 ,k2 ,… ,ks 是适合 k1 +k2 +… +ks=n的非负整数 ,正整数 nk1 k2 … ks=n!k1 !k2 !… ks!称为多项式系数 .本文讨论了当n=a0 +a1 p+a2 p2 +… +arpr ,其中 p为素数且 p≤ n,0≤ ai<p(0≤i≤r) ;ki=a( i)0 +a( i)1 p+… +a( i)r pr,其中 ki≥ 0 ,∑si=1ki=n,0≤a( i)k <p(0≤i<s)时多项式系数的整除性问题 ,得出的结果推广了著名的 Lucas定理[1 ] .

Suppose n is a positive integer and k1,k2 , ...,k s are nonnegative numbers with k 1+k 2+...+k s=n, positive integer nk1k2...ks=n!k1 !k 2!...k s! be called multinomial number. In this paper, we discuss its divisibility when n=a 0+a 1p+a 2p 2+...+a rpr and p is a prime number and p≤n,0≤a i<p (0≤i≤r); k i=a (i) 0+a (i ) 1p+...+a (i) rp r, and k i≥0,HZDDsi=1 k i=n,0≤a (i) k<p (0≤i≤s) and our result generalized Lucas theorem.