Zeilberger算法、Petkovek算法与一类组合和是否有闭形式的问题

ZEILBERGER'S ALGORITHM,PETKOVEK'S ALGORITHM AND THE PROBLEM WHETHER A KIND OF COMBINATORIAL SUM HAS A CLOSED FORM

提出研究形如fp,r,s,x(n)=∑ from k=0 to rn 〔pn k〕s xk的组合和的闭形式问题的Z-P方法,并利用此方法得到了如下结果:1)当s=1,p=2r时证明了对未定元x,∑ from k=0 to rn 〔2rn k〕xk无闭形式表示;2)对p、r、s及x取特定的值,借助计算机归纳出几个值得探索的猜测.

The Z - P method is given to study the problem whether a kind of combinatorial fp,r,s,x（n）=^rn∑k=0（pn k）^s x^k has a closed form. By using this method, some results are obmined： 1 ）when s = 1 ,p = 2r, it＇s proved that for an indeterminate x,^rn∑ k=0（2rn k）x^k has no closed form; 2） when p, r, s and x are given with some specific values, several conjectures, are obtained inductively,with the help of computer.