This function is an implementation of Koepf's extension of Zeilberger's algorithm, calculating a (downward) recurrence equation for the sum

the sum to be taken over all integers k, with respect to n if f is an (m,l)-fold hypergeometric term with respect to (n,k) for some m and l. The minimal values for m, and l are determined automatically.

The output is a recurrence which equals zero. The recurrence is a function of n the recurrence variable and .

An expression f is called (m,l)-fold hypergeometric term with respect to (n,k) if

are rational with respect to n and k. This is typically the case for ratios of products of rational functions, exponentials, factorials, binomial coefficients, and Pochhammer symbols that are rational-linear in their arguments. The implementation supports this type of input.

The command with(sumtools,sumrecursion) allows the use of the abbreviated form of this command.