  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, $\Gamma$ 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, $\Gamma$ 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 integer-linear hypergeometric identities is extended to
rational-linear hypergeometric identities.
