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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 Tn,k of n and k, MinimalZpair(T, n, k, En) constructs for Tn,k the minimal Z-pair L,G; MinimalTelescoper(T, n, k, En) constructs for Tn,k the minimal telescoper L.

• 

L and G satisfy the following properties:

  

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

  

2. G is a hypergeometric term of n and k.

  

3. LT=Ek1G, where Ek 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 Tn,k.

  

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

  

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

  

b. If Tn,k is a nonrational term, first compute a lower bound u for the order of the telescopers for Tn,k. 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 T1,T2 of T is ``simpler'' than T in some sense, we first apply Zeilberger's algorithm to T2 to obtain the minimal Z-pair L,G for T2. It is easy to show that L,LT1+G is the minimal Z-pair for the input term T.

Examples

withSumToolsHypergeometric:

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

T1k1nk+1binomialn+1,kbinomial2n2k1,n1

T−1kn+1k2n2k1n1nk+1

(1)

MinimalZpairT,n,k,En

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

Case 2a: Rational Function

T13n+20k+23

T13n+20k+23

(2)

MinimalZpairT,n,k,En

En201,13n+42+20k3+13n+20k+223+13n+20k+23

(3)

Case 2b: Hypergeometric

T1nk+11n2k42n+k+4!1nk1n2k22n+k+3!+1n2k22n+k+3!

T1nk+11n42k2n+k+4!1nk1n2k22n+k+3!+1n2k22n+k+3!

(4)

ZpairMinimalZpairT,n,k,En:

Zpair1

1953125n944140625n8438125000n72505718750n69095640625n521719685625n434096450250n333905768600n219362572120n4833216960En3+1953125n9+42187500n8+400625000n7+2194468750n6+7637609375n5+17505613750n4+26405971500n3+25257742600n2+13888257120n+3340995840En2+20000n4+152000n3+422400n2+508160n+223232En20000n4232000n3998400n21888960n1325792

(5)

T1n2+9nk4n22k2+21k5

T122k2+9nk+n2+21k4n5

(6)

MinimalTelescoperT,n,k,En

13n1+1413nEn+144+13nEn11+157+13nEn12

(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]