IsZApplicable - Maple Help

SumTools[Hypergeometric]

 IsZApplicable
 test the applicability of Zeilberger's algorithm to hypergeometric terms

 Calling Sequence IsZApplicable(F, n, k, En, 'Zpair')

Parameters

 F - hypergeometric term in n and k n - name k - name En - (optional) name; denote the shift operator with respect to n 'Zpair' - (optional) name; assigned computed Z-pair

Description

 • Let F be a hypergeometric term in n and k. The IsZApplicable(F, n, k, En, 'Zpair') command determines the applicability of Zeilberger's algorithm to F. It returns true if Zeilberger's algorithm is applicable to F. Otherwise, it returns false.
 • If Zeilberger's algorithm is applicable to the function F and the fourth and the fifth optional arguments are specified, the fifth argument 'Zpair' is assigned the computed Z-pair $L,G$ for F.
 • If the input F is not a rational function of n and k, IsZApplicable returns FAIL. In this case, if the optional arguments En and 'Zpair' are specified, Zeilberger(F, n, k, En) is called. If it succeeds in finding a Z-pair for F, the computed Z-pair is assigned to 'Zpair'.

Examples

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

In the following example, F is not a proper hypergeometric term. However, Zeilberger's algorithm is applicable to F:

 > $F≔\frac{1}{nk+n+1}\mathrm{binomial}\left(n,k+1\right)+\frac{nk-2k-n+2}{{n}^{2}k+2n{k}^{2}-nk+2k+n-1}\mathrm{binomial}\left(n,k\right)$
 ${F}{≔}\frac{\left(\genfrac{}{}{0}{}{{n}}{{k}{+}{1}}\right)}{{n}{}{k}{+}{n}{+}{1}}{+}\frac{\left({n}{}{k}{-}{2}{}{k}{-}{n}{+}{2}\right){}\left(\genfrac{}{}{0}{}{{n}}{{k}}\right)}{{2}{}{n}{}{{k}}^{{2}}{+}{{n}}^{{2}}{}{k}{-}{n}{}{k}{+}{2}{}{k}{+}{n}{-}{1}}$ (1)
 > $\mathrm{IsProperHypergeometricTerm}\left(F,n,k\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{IsZApplicable}\left(F,n,k\right)$
 ${\mathrm{true}}$ (3)

In the following example, F is not a proper hypergeometric term, and Zeilberger's algorithm is not applicable to F either:

 > $F≔\frac{{\left(-1\right)}^{k}\cdot 1}{nk+1}\mathrm{binomial}\left(n+1,k\right)\mathrm{binomial}\left(2n-2k-1,n-1\right)$
 ${F}{≔}\frac{{\left({-1}\right)}^{{k}}{}\left(\genfrac{}{}{0}{}{{n}{+}{1}}{{k}}\right){}\left(\genfrac{}{}{0}{}{{2}{}{n}{-}{2}{}{k}{-}{1}}{{n}{-}{1}}\right)}{{n}{}{k}{+}{1}}$ (4)
 > $\mathrm{IsProperHypergeometricTerm}\left(F,n,k\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{IsZApplicable}\left(F,n,k\right)$
 ${\mathrm{false}}$ (6)

The input is a hypergeometric term of n and k.

 > $F≔\frac{1}{{n}^{2}+3nk-2n-10{k}^{2}+11k-3}$
 ${F}{≔}\frac{{1}}{{-}{10}{}{{k}}^{{2}}{+}{3}{}{n}{}{k}{+}{{n}}^{{2}}{+}{11}{}{k}{-}{2}{}{n}{-}{3}}$ (7)
 > $\mathrm{IsZApplicable}\left(F,n,k,\mathrm{En},'\mathrm{Zpair}'\right)$
 ${\mathrm{true}}$ (8)
 > $L≔\mathrm{Zpair}\left[1\right]$
 ${L}{≔}\left({-}{7}{}{n}{-}{41}\right){}{{\mathrm{En}}}^{{6}}{+}\left({-}{7}{}{n}{-}{34}\right){}{{\mathrm{En}}}^{{5}}{+}{\mathrm{En}}{}\left({7}{}{n}{+}{6}\right){-}{1}{+}{7}{}{n}$ (9)
 > $G≔\mathrm{Zpair}\left[2\right]$
 ${G}{≔}\frac{\left(\frac{{5600}{}{{k}}^{{6}}}{{7}{}{n}{+}{6}}{-}\frac{{1440}{}\left({7}{}{n}{+}{45}\right){}{{k}}^{{5}}}{{7}{}{n}{+}{6}}{+}\frac{{40}{}\left({154}{}{{n}}^{{2}}{+}{2230}{}{n}{+}{7399}\right){}{{k}}^{{4}}}{{7}{}{n}{+}{6}}{-}\frac{{80}{}\left({14}{}{{n}}^{{3}}{+}{450}{}{{n}}^{{2}}{+}{3619}{}{n}{+}{8445}\right){}{{k}}^{{3}}}{{7}{}{n}{+}{6}}{-}\frac{{30}{}\left({7}{}{{n}}^{{4}}{-}{40}{}{{n}}^{{3}}{-}{2079}{}{{n}}^{{2}}{-}{13900}{}{n}{-}{26696}\right){}{{k}}^{{2}}}{{7}{}{n}{+}{6}}{+}\frac{{10}{}\left({7}{}{{n}}^{{5}}{+}{145}{}{{n}}^{{4}}{+}{679}{}{{n}}^{{3}}{-}{2545}{}{{n}}^{{2}}{-}{25082}{}{n}{-}{45780}\right){}{k}}{{7}{}{n}{+}{6}}{-}\frac{{75}{}{{n}}^{{5}}{+}{1295}{}{{n}}^{{4}}{+}{7075}{}{{n}}^{{3}}{+}{7337}{}{{n}}^{{2}}{-}{44310}{}{n}{-}{96048}}{{7}{}{n}{+}{6}}\right){}\left({10}{}{\left({k}{-}{1}\right)}^{{2}}{-}{3}{}{n}{}\left({k}{-}{1}\right){-}{{n}}^{{2}}{+}{9}{}{k}{-}{7}{-}{n}\right){}\left({7}{}{n}{+}{6}\right)}{\left({5}{}{k}{+}{n}{-}{3}\right){}\left({5}{}{k}{+}{n}{-}{2}\right){}\left({2}{}{k}{-}{n}{-}{3}\right){}\left({-}{4}{+}{2}{}{k}{-}{n}\right){}\left({-}{5}{+}{2}{}{k}{-}{n}\right){}\left({-}{6}{+}{2}{}{k}{-}{n}\right){}\left({-}{7}{+}{2}{}{k}{-}{n}\right){}\left({-}{10}{}{{k}}^{{2}}{+}{3}{}{n}{}{k}{+}{{n}}^{{2}}{+}{11}{}{k}{-}{2}{}{n}{-}{3}\right)}$ (10)
 > $\mathrm{Verify}\left(F,\mathrm{Zpair},n,k,\mathrm{En}\right)$
 ${\mathrm{true}}$ (11)

References

 Abramov, S.A. "Applicability of Zeilberger's Algorithm to Hypergeometric Terms." Proceedings ISSAC'2002, pp. 1-7. ACM Press, 2002.
 Abramov, S.A., and Le, H.Q. "Applicability of Zeilberger's Algorithm to Rational Functions." Proceedings FPSAC'2000, pp. 91-102. Springer-Verlag LNCS, 2000.