SumTools[Hypergeometric] - Maple Programming Help

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SumTools[Hypergeometric]

 LowerBound
 compute a lower bound for the order of the telescopers for a hypergeometric term

 Calling Sequence LowerBound(T, n, k, En, 'Zpair')

Parameters

 T - hypergeometric term in n and k n - name k - name En - (optional) name denoting the shift operator with respect to n 'Zpair' - (optional) name

Description

 • Let T be a hypergeometric term in n and k. The function LowerBound(T, n, k) computes a lower bound for the order of the telescopers for T if it is guaranteed that Zeilberger's algorithm is applicable to T.
 • If the fourth and the fifth optional arguments (each of which can be any name), En and 'Zpair' respectively, are specified, the minimal telescoper for T is computed and assigned to the fifth argument 'Zpair' using the computed lower bound  as the starting value of the guessed orders.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right):$
 > $T≔\frac{1}{n\left(k+1\right)+1}\mathrm{binomial}\left(2n,k+1\right)-\frac{1}{nk+1}\mathrm{binomial}\left(2n,k\right)+\frac{1}{\left(2k-1\right)\left(n-3k+1\right)}\mathrm{binomial}\left(2n,k\right)$
 ${T}{≔}\frac{\left(\genfrac{}{}{0}{}{{2}{}{n}}{{k}{+}{1}}\right)}{{n}{}\left({k}{+}{1}\right){+}{1}}{-}\frac{\left(\genfrac{}{}{0}{}{{2}{}{n}}{{k}}\right)}{{n}{}{k}{+}{1}}{+}\frac{\left(\genfrac{}{}{0}{}{{2}{}{n}}{{k}}\right)}{\left({2}{}{k}{-}{1}\right){}\left({n}{-}{3}{}{k}{+}{1}\right)}$ (1)
 > $\mathrm{LowerBound}\left(T,n,k\right)$
 ${3}$ (2)

Zeilberger's algorithm is not applicable to the following hypergeometric term so LowerBound returns an error.

 > $T≔\frac{1}{nk+1}\mathrm{binomial}\left(2n,2k\right)$
 ${T}{≔}\frac{\left(\genfrac{}{}{0}{}{{2}{}{n}}{{2}{}{k}}\right)}{{n}{}{k}{+}{1}}$ (3)
 > $\mathrm{LowerBound}\left(T,n,k\right)$
 > $T≔\frac{1}{\left(n\left(k+1\right)-1\right)\left(n-3k-5\right)\left(2n+k+4\right)!}-\frac{1}{\left(nk-1\right)\left(n-3k-2\right)\left(2n+k+3\right)!}+\frac{1}{\left(n-3k-2\right)\left(2n+k+3\right)!}$
 ${T}{≔}\frac{{1}}{\left({n}{}\left({k}{+}{1}\right){-}{1}\right){}\left({n}{-}{3}{}{k}{-}{5}\right){}\left({2}{}{n}{+}{k}{+}{4}\right){!}}{-}\frac{{1}}{\left({n}{}{k}{-}{1}\right){}\left({n}{-}{3}{}{k}{-}{2}\right){}\left({2}{}{n}{+}{k}{+}{3}\right){!}}{+}\frac{{1}}{\left({n}{-}{3}{}{k}{-}{2}\right){}\left({2}{}{n}{+}{k}{+}{3}\right){!}}$ (4)
 > $\mathrm{LowerBound}\left(T,n,k,\mathrm{En},'\mathrm{Zpair}'\right)$
 ${3}$ (5)
 > $L≔\mathrm{Zpair}\left[1\right]$
 ${L}{≔}\left({-}{96889010407}{}{{n}}^{{13}}{-}{4013973288290}{}{{n}}^{{12}}{-}{76107306338070}{}{{n}}^{{11}}{-}{874305244269093}{}{{n}}^{{10}}{-}{6788048750132832}{}{{n}}^{{9}}{-}{37604322096371100}{}{{n}}^{{8}}{-}{152885294205849709}{}{{n}}^{{7}}{-}{461743890026242439}{}{{n}}^{{6}}{-}{1035633823402072251}{}{{n}}^{{5}}{-}{1703061496353656040}{}{{n}}^{{4}}{-}{1995094474254403011}{}{{n}}^{{3}}{-}{1575944956962320238}{}{{n}}^{{2}}{-}{751943328788699320}{}{n}{-}{163575961093126400}\right){}{{\mathrm{En}}}^{{4}}{+}\left({96889010407}{}{{n}}^{{13}}{+}{3917084277883}{}{{n}}^{{12}}{+}{72536254240212}{}{{n}}^{{11}}{+}{814487155639857}{}{{n}}^{{10}}{+}{6186007839562887}{}{{n}}^{{9}}{+}{33550538764167390}{}{{n}}^{{8}}{+}{133652029105976437}{}{{n}}^{{7}}{+}{395832377416110838}{}{{n}}^{{6}}{+}{871303942188476181}{}{{n}}^{{5}}{+}{1407347883183343752}{}{{n}}^{{4}}{+}{1620685980982353516}{}{{n}}^{{3}}{+}{1259506839996666240}{}{{n}}^{{2}}{+}{591742636413140800}{}{n}{+}{126860211237760000}\right){}{{\mathrm{En}}}^{{3}}{+}\left({257298363}{}{{n}}^{{6}}{+}{3969746172}{}{{n}}^{{5}}{+}{25015702068}{}{{n}}^{{4}}{+}{82342227429}{}{{n}}^{{3}}{+}{149184720027}{}{{n}}^{{2}}{+}{140923968318}{}{n}{+}{54171659763}\right){}{\mathrm{En}}{-}{257298363}{}{{n}}^{{6}}{-}{5513536350}{}{{n}}^{{5}}{-}{48723908373}{}{{n}}^{{4}}{-}{227248464681}{}{{n}}^{{3}}{-}{589862551887}{}{{n}}^{{2}}{-}{807775419969}{}{n}{-}{455865322140}$ (6)

The computed lower bound is 3, while the order of the minimal telescoper is

 > $\mathrm{degree}\left(L,\mathrm{En}\right)$
 ${4}$ (7)

References

 Abramov, S.A. and Le, H.Q. "A Lower Bound for the Order of Telescopers for a Hypergeometric Term." CD-ROM. Proceedings FPSAC 2002. (2002).