 SumTools[Hypergeometric] - Maple Programming Help

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SumTools[Hypergeometric]

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

 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 $\mathrm{En}\left(F\left(n,k\right)\right)=F\left(n+1,k\right)$. The ZpairDirect(F, n, k, En) command computes a Z-pair $L,G$ such that

$LF\left(n,k\right)=G\left(n,k+1\right)-G\left(n,k\right).$

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right):$
 > $F≔\frac{1}{{\left(3n+20k+2\right)}^{3}}$
 ${F}{≔}\frac{{1}}{{\left({3}{}{n}{+}{20}{}{k}{+}{2}\right)}^{{3}}}$ (1)

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

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

 > $F≔\frac{1}{{k}^{5}+{k}^{3}n+3{k}^{3}-5n{k}^{2}-2{k}^{2}-5{n}^{2}-17n-6}$
 ${F}{≔}\frac{{1}}{{{k}}^{{5}}{+}{{k}}^{{3}}{}{n}{+}{3}{}{{k}}^{{3}}{-}{5}{}{n}{}{{k}}^{{2}}{-}{2}{}{{k}}^{{2}}{-}{5}{}{{n}}^{{2}}{-}{17}{}{n}{-}{6}}$ (3)
 > $\mathrm{infolevel}\left[\mathrm{ZpairDirect}\right]≔0:$
 > $\mathrm{ZpairDirect}\left(F,n,k,\mathrm{En}\right)$

References

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