不定求和与定求和的算法
Algorithms for the indefinite and definite summation
摘要：
著名的Zeilberger算法，找到完整的递归方程的超几何项$F(n,k)$的定和推广到某些非超几何项。如果$F(n+1,k)/F(n,k)$和$F(n,k+1)/F(n,k)$都是有理函数，则表达式$F(n,k)$称为超几何项。典型的例子是指数、因子、$Γ$函数项、二项式系数和Pochhammer符号的乘积的比率，它们在自变量中相对于$n$和$k$是整数线性的。我们考虑更一般的情况下的指数，因子，$Γ$函数项，二项式系数，和Pochhammer符号的产品的比率是合理的线性相对于$n$和$k$在他们的参数，并提出了一个扩展版本的Zeilberger算法的这种情况下，使用一个扩展版本的Goalster算法的不定求和。以类似的方式，将整数线性超几何恒等式的有理函数证明的Wilf-Zeilberger方法推广到有理线性超几何恒等式。
英文摘要：
The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms $F(n,k)$ is extended to certain nonhypergeometric terms. An expression $F(n,k)$ is called a hypergeometric term if both $F(n+1,k)/F(n,k)$ and $F(n,k+1)/F(n,k)$ are rational functions. Typical examples are ratios of products of exponentials, factorials, $Γ$ function terms, bin omial coefficients, and Pochhammer symbols that are integer-linear with respect to $n$ and $k$ in their arguments. We consider the more general case of ratios of products of exponentials, factorials, $Γ$ function terms, binomial coefficients, and Pochhammer symbols that are rational-linear with respect to $n$ and $k$ in their arguments, and present an extended version of Zeilberger's algorithm for this case, using an extended version of Gosper's algorithm for indefinite summation. In a similar way the Wilf-Zeilberger method of rational function certification of integer-linear hypergeometric identities is extended to rational-linear hypergeometric identities.
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