Let F be a rational function of n and k, En the shift operator with respect to n defined by $\mathrm{En}\left(F\left(n\,k\right)\right)=F\left(n+1\,k\right)$. 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 $\left[L\,G\right]$ 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.

$\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right)\:$

$F≔\frac{1}{{\left(3n+20k+2\right)}^3}$

$F≔\frac{1}{{\left(3n+20k+2\right)}^3}$

$\mathrm{infolevel}\left[\mathrm{ZpairDirect}\right]≔3\:$

$\mathrm{ZpairDirect}\left(F\,n\,k\,\mathrm{En}\right)$

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"

$\left[{\mathrm{En}}^{20}-1\,\frac{1}{{\left(3n+20k+42\right)}^3}+\frac{1}{{\left(3n+20k+22\right)}^3}+\frac{1}{{\left(3n+20k+2\right)}^3}\right]$

$F≔\frac{1}{k^5+k^{3}n+3k^3-5n{k}^2-2k^2-5n^2-17n-6}$

$F≔\frac{1}{k^5+k^{3}n+3k^3-5n{k}^2-2k^2-5n^2-17n-6}$

$\mathrm{infolevel}\left[\mathrm{ZpairDirect}\right]≔0\:$

$\mathrm{ZpairDirect}\left(F\,n\,k\,\mathrm{En}\right)$

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.