Algorithms for the Indefinite and Definite Summation

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 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, \Gamma function terms, binomial 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, \Gamma 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 integerlinear hypergeometric i..