perform direct algorithm to construct Zeilberger's recurrences for rational functions
ZpairDirect(F, n, k, En)
rational function of n and k
name; denote the shift operator with respect to n
Let F be a rational function of n and k, En the shift operator with respect to n defined by En⁡F⁡n,k=F⁡n+1,k. The ZpairDirect(F, n, k, En) command computes a Z-pair L,G such that
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.
F ≔ 13⁢n+20⁢k+23
Set the infolevel to 3.
infolevelZpairDirect ≔ 3:
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"
If the routine cannot determine a Z-pair, Maple returns an error.
F ≔ 1k5+k3⁢n+3⁢k3−5⁢n⁢k2−2⁢k2−5⁢n2−17⁢n−6
infolevelZpairDirect ≔ 0:
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)
Le, H.Q. "A Direct Algorithm to Construct Zeilberger's Recurrences for Rational Functions." Proceedings FPSAC'2001, pp. 303-312. 2001.
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