For a specified hypergeometric term $T\left(n\,k\right)$ of n and k, MinimalZpair(T, n, k, En) constructs for $T\left(n\,k\right)$ the minimal Z-pair $\left[L\,G\right]$; MinimalTelescoper(T, n, k, En) constructs for $T\left(n\,k\right)$ the minimal telescoper $L$.

L and G satisfy the following properties:

1. $L$ is a linear recurrence operator in En with polynomial coefficients in n.

2. $G$ is a hypergeometric term of n and k.

3. $LT=\left(\mathrm{Ek}-1\right)G$, where $\mathrm{Ek}$ denotes the shift operator with respect to k.

4. The order of L w.r.t. En is minimal.

The execution steps of MinimalZpair can be described as follows.

1. Determine the applicability of Zeilberger's algorithm to $T\left(n\,k\right)$.

2. If it is proven in Step 1 that a Z-pair for $T\left(n\,k\right)$ does not exist, return the conclusive error message ``Zeilberger's algorithm is not applicable''. Otherwise,

a. If $T\left(n\,k\right)$ is a rational function in n and k, apply the direct algorithm to compute the minimal Z-pair for $T\left(n\,k\right)$.

b. If $T\left(n\,k\right)$ is a nonrational term, first compute a lower bound u for the order of the telescopers for $T\left(n\,k\right)$. Then compute the minimal Z-pair using Zeilberger's algorithm with u as the starting value for the guessed orders.

For case 2b, since the term T2 in the additive decomposition $\mathrm{T1},\mathrm{T2}$ of T is ``simpler'' than T in some sense, we first apply Zeilberger's algorithm to T2 to obtain the minimal Z-pair $\left[L\,G\right]$ for T2. It is easy to show that $\left[L\,\mathrm{LT1}+G\right]$ is the minimal Z-pair for the input term T.

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{Zpair}≔\mathrm{MinimalZpair}\left(T\,n\,k\,\mathrm{En}\right)\:$

$\mathrm{Zpair}\left[1\right]$

$\left(-1953125n^9-44140625n^8-438125000n^7-2505718750n^6-9095640625n^5-21719685625n^4-34096450250n^3-33905768600n^2-19362572120n-4833216960\right){\mathrm{En}}^3+\left(1953125n^9+42187500n^8+400625000n^7+2194468750n^6+7637609375n^5+17505613750n^4+26405971500n^3+25257742600n^2+13888257120n+3340995840\right){\mathrm{En}}^2+\left(20000n^4+152000n^3+422400n^2+508160n+223232\right)\mathrm{En}-20000n^4-232000n^3-998400n^2-1888960n-1325792$

$T≔\frac{1}{n^2+9nk-4n-22k^2+21k-5}$

$T≔\frac{1}{-22k^2+9kn+n^2+21k-4n-5}$

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

$-13n-1+\left(-14-13n\right)\mathrm{En}+\left(144+13n\right){\mathrm{En}}^{11}+\left(157+13n\right){\mathrm{En}}^{12}$

Abramov, S.A.; Geddes, K.O.; and Le, H.Q. "Computer Algebra Library for the Construction of the Minimal Telescopers." Proceedings ICMS'2002, pp. 319- 329. World Scientific, 2002.