MinimalTelescoper - Maple Help
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SumTools[Hypergeometric]

  

MinimalZpair

  

compute the minimal Z-pair

  

MinimalTelescoper

  

compute the minimal telescoper

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

MinimalZpair(T, n, k, En)

MinimalTelescoper(T, n, k, En)

Parameters

T

-

hypergeometric term of n and k

n

-

name

k

-

name

En

-

name; denote the shift operator with respect to n

Description

• 

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.

Examples

Case 1: Zeilberger's algorithm is not applicable to the input term T.

(1)

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

Case 2a: Rational Function

(2)

(3)

Case 2b: Hypergeometric

(4)

(5)

(6)

(7)

References

  

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]

SumTools[Hypergeometric][IsZApplicable]

SumTools[Hypergeometric][LowerBound]

SumTools[Hypergeometric][Zeilberger]

SumTools[Hypergeometric][ZpairDirect]

 


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