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SumTools[Hypergeometric]

  

ZpairDirect

  

perform direct algorithm to construct Zeilberger's recurrences for rational functions

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

ZpairDirect(F, n, k, En)

Parameters

F

-

rational function of n and k

n

-

name

k

-

name

En

-

name; denote the shift operator with respect to n

Description

• 

Let F be a rational function of n and k, En the shift operator with respect to n defined by EnFn,k=Fn+1,k. The ZpairDirect(F, n, k, En) command computes a Z-pair L,G such that

LFn,k=Gn,k+1Gn,k.

• 

The output from ZpairDirect is a list of two elements L,G representing the computed Z-pair L,G provided such a pair exists.

• 

The main distinction between ZpairDirect and Zeilberger's algorithm is that Zeilberger's algorithm uses an item-by-item examination technique for the order of the computed difference operator L.  For more information, see Zeilberger.

  

The function ZpairDirect, on the other hand, uses a direct algorithm to construct a Z-pair L,G for F. It first determines if there exists a Z-pair for F. If the answer is positive, it computes a Z-pair directly. Otherwise, it gives the conclusive error message ``there does not exist a Z-pair for F'' where F is the input rational function. When the Zeilberger routine is used, and if the input hypergeometric term T is also a rational function, ZpairDirect is invoked.

• 

For the ZpairDirect routine, the input F must be a rational function.

  

Note: If you set infolevel[ZpairDirect] to 3, Maple prints diagnostics.

Examples

withSumToolsHypergeometric:

F13n+20k+23

F13n+20k+23

(1)

Set the infolevel to 3.

infolevelZpairDirect3:

ZpairDirectF,n,k,En

ZpairDirect:   "Check for the existence of a Z-pair"
ZpairDirect:   "There exists a Z-pair"
ZpairDirect:   "Start computing a Z-pair for the given rational function"

En201,13n+20k+423+13n+20k+223+13n+20k+23

(2)

If the routine cannot determine a Z-pair, Maple returns an error.

F1k5+k3n+3k35nk22k25n217n6

F1k5+k3n+3k35nk22k25n217n6

(3)

infolevelZpairDirect0:

ZpairDirectF,n,k,En

Error, (in SumTools:-Hypergeometric:-ZpairDirect) there does not exist a Z-pair for 1/(k^5+k^3*n+3*k^3-5*k^2*n-2*k^2-5*n^2-17*n-6)

References

  

Le, H.Q. "A Direct Algorithm to Construct Zeilberger's Recurrences for Rational Functions." Proceedings FPSAC'2001, pp. 303-312. 2001.

See Also

infolevel

SumTools[Hypergeometric]

SumTools[Hypergeometric][IsZApplicable]

SumTools[Hypergeometric][MinimalZpair]

SumTools[Hypergeometric][Zeilberger]