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.

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

$T≔\frac{1}{n\left(k+1\right)+1}\mathrm{binomial}\left(2n\,k+1\right)-\frac{1}{nk+1}\mathrm{binomial}\left(2n\,k\right)+\frac{1}{\left(2k-1\right)\left(n-3k+1\right)}\mathrm{binomial}\left(2n\,k\right)$

$T≔\frac{\left(\genfrac{}{}{0ex}{}{2n}{k+1}\right)}{n\left(k+1\right)+1}-\frac{\left(\genfrac{}{}{0ex}{}{2n}{k}\right)}{nk+1}+\frac{\left(\genfrac{}{}{0ex}{}{2n}{k}\right)}{\left(2k-1\right)\left(n-3k+1\right)}$

$\mathrm{LowerBound}\left(T\,n\,k\right)$

$3$

$T≔\frac{1}{nk+1}\mathrm{binomial}\left(2n\,2k\right)$

$T≔\frac{\left(\genfrac{}{}{0ex}{}{2n}{2k}\right)}{nk+1}$

$\mathrm{LowerBound}\left(T\,n\,k\right)$

Error, (in SumTools:-Hypergeometric:-LowerBound) Zeilberger's algorithm is not applicable

$T≔\frac{1}{\left(n\left(k+1\right)-1\right)\left(n-3k-5\right)\left(2n+k+4\right)!}-\frac{1}{\left(nk-1\right)\left(n-3k-2\right)\left(2n+k+3\right)!}+\frac{1}{\left(n-3k-2\right)\left(2n+k+3\right)!}$

$T≔\frac{1}{\left(n\left(k+1\right)-1\right)\left(n-3k-5\right)\left(2n+k+4\right)!}-\frac{1}{\left(nk-1\right)\left(n-3k-2\right)\left(2n+k+3\right)!}+\frac{1}{\left(n-3k-2\right)\left(2n+k+3\right)!}$

$\mathrm{LowerBound}\left(T\,n\,k\,\mathrm{En}\,'\mathrm{Zpair}'\right)$

$3$

$L≔\mathrm{Zpair}\left[1\right]$

$L≔\left(-96889010407n^{13}-4013973288290n^{12}-76107306338070n^{11}-874305244269093n^{10}-6788048750132832n^9-37604322096371100n^8-152885294205849709n^7-461743890026242439n^6-1035633823402072251n^5-1703061496353656040n^4-1995094474254403011n^3-1575944956962320238n^2-751943328788699320n-163575961093126400\right){\mathrm{En}}^4+\left(96889010407n^{13}+3917084277883n^{12}+72536254240212n^{11}+814487155639857n^{10}+6186007839562887n^9+33550538764167390n^8+133652029105976437n^7+395832377416110838n^6+871303942188476181n^5+1407347883183343752n^4+1620685980982353516n^3+1259506839996666240n^2+591742636413140800n+126860211237760000\right){\mathrm{En}}^3+\left(257298363n^6+3969746172n^5+25015702068n^4+82342227429n^3+149184720027n^2+140923968318n+54171659763\right)\mathrm{En}-257298363n^6-5513536350n^5-48723908373n^4-227248464681n^3-589862551887n^2-807775419969n-455865322140$

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

$4$

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).