SumTools[Hypergeometric]
MinimalZpair
compute the minimal Z-pair
MinimalTelescoper
compute the minimal telescoper
Calling Sequence
Parameters
Description
Examples
References
MinimalZpair(T, n, k, En)
MinimalTelescoper(T, n, k, En)
T
-
hypergeometric term of n and k
n
name
k
En
name; denote the shift operator with respect to n
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.
Case 1: Zeilberger's algorithm is not applicable to the input term T.
Error, (in SumTools:-Hypergeometric:-MinimalZpair) Zeilberger's algorithm is not applicable
Case 2a: Rational Function
Case 2b: Hypergeometric
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.
See Also
SumTools[Hypergeometric][IsZApplicable]
SumTools[Hypergeometric][LowerBound]
SumTools[Hypergeometric][Zeilberger]
SumTools[Hypergeometric][ZpairDirect]
Download Help Document