1994/12/17 by Koepf, Wolfram
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/9412227
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.