For a specified hypergeometric term  of n and k, MinimalZpair(T, n, k, En) constructs for  the minimal Z-pair ; MinimalTelescoper(T, n, k, En) constructs for  the minimal telescoper .

L and G satisfy the following properties:

1.  is a linear recurrence operator in En with polynomial coefficients in n.

2.  is a hypergeometric term of n and k.

3. , where  denotes the shift operator with respect to k.

4. The order of L w.r.t. En is minimal.

The execution steps of MinimalZpair can be described as follows.

1. Determine the applicability of Zeilberger's algorithm to .

2. If it is proven in Step 1 that a Z-pair for  does not exist, return the conclusive error message ``Zeilberger's algorithm is not applicable''. Otherwise,

a. If  is a rational function in n and k, apply the direct algorithm to compute the minimal Z-pair for .

b. If  is a nonrational term, first compute a lower bound u for the order of the telescopers for . Then compute the minimal Z-pair using Zeilberger's algorithm with u as the starting value for the guessed orders.

For case 2b, since the term T2 in the additive decomposition  of T is ``simpler'' than T in some sense, we first apply Zeilberger's algorithm to T2 to obtain the minimal Z-pair  for T2. It is easy to show that  is the minimal Z-pair for the input term T.

Error, (in SumTools:-Hypergeometric:-MinimalZpair) Zeilberger's algorithm is not applicable

Abramov, S.A.; Geddes, K.O.; and Le, H.Q. "Computer Algebra Library for the Construction of the Minimal Telescopers." Proceedings ICMS'2002, pp. 319- 329. World Scientific, 2002.