For a specified hypergeometric term  of n and k, the Zeilberger(T, n, k, En) command constructs for  a Z-pair  that consists of a linear difference operator with coefficients that are polynomials of n over the complex number field

and a function  such that

En is the shift operator with respect to n, defined by .

The algorithm uses an item-by-item examination of the order v of L. It starts with the value of 0 for v and increases v until it is successful in finding a Z-pair  provided that such a pair exists. Note that a maximum value of the order of the difference operator L in the Z-pair  must be specified. This can be controlled by assigning a value to the global variable _MAXORDER (the default value of _MAXORDER is ). Additionally, by assigning values to the global variables _MINORDER and _MAXORDER, the algorithm is restricted to finding a Z-pair  for  such that the order of L is between _MINORDER and _MAXORDER.

The algorithm has two implementations. The default implementation is based on the universal denominators, and another one uses a variant of Gosper's algorithm. With the 'gosper' option, Gosper-based implementation is used.

The output from the Zeilberger command is a list of two elements  representing the computed Z-pair .

The ZeilbergerRecurrence(T, n, k, f, l..u) command constructs the Zeilberger's recurrence. The function first computes the Z-pair  for T and sums both sides of the relation  over k in the specified range . This yields in general an inhomogeneous recurrence equation.

The possible ranges of  include , , , and -infinity..infinity where r, s, u, and v are integers.

The Verify(T, 'Zpair', n, k, En) command verifies the correctness of the result ('Zpair') returned from Zeilberger, that is, it tries to verify that:

Error, (in SumTools:-Hypergeometric:-Zeilberger) No recurrence of order 6 was found

Petkovsek, M.; Wilf, H.; and Zeilberger, D. A=B. Wellesley, Massachusetts: A K Peters, Ltd., 1996.

Zeilberger, D. "The Method of Creative Telescoping." Journal of Symbolic Computing. Vol. 11. (1991): 195-204.