The following optional arguments can be used if $T$ is a rational function of $n$.

minimize=v, where v is either a numeric value between $0$ and $1$ or one of "numerator", "sum denominator", "combined", "left", "right".

If v="sum denominator", then the degree $g$ of the denominator of $T_1$ will be minimized.

If v="numerator", then the degree $e$ of the numerator of $T_2$ will be minimized.

If v="combined", then then the sum of the degrees $g+e$ will be  minimized.

If v is  a numeric constant between $0$ and $1$, then the weighted sum of the degrees $vg+\left(1-v\right)e$ will be minimized.

Note that small values of v may lead to time-consuming search; the option maxiterations (see below) can be used to restrict it.

If v="left", then the remainder of the result will be aligned such that $\gcd \left(\mathrm{denom}\left(T_1\right)\&comma;\genfrac{}{}{0ex}{}{\mathrm{denom}\left(T_2\right)}{\phantom{n=n+k}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{denom}\left(T_2\right)}}{n=n+k}\right)=1$ for all integers $k<0$.

If v="right", then the remainder of the result will be aligned such that $\gcd \left(\mathrm{denom}\left(T_1\right)\&comma;\genfrac{}{}{0ex}{}{\mathrm{denom}\left(T_2\right)}{\phantom{n=n+k}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{denom}\left(T_2\right)}}{n=n+k}\right)=1$ for all integers $0\le k$.

maxiterations=integer

This option can be used to restrict the number of iterations performed by the command when the option minimize=v is used with a small positive numeric value v. The default value is $10000$.

The SumDecomposition(T, n, k) command constructs two hypergeometric terms $T_1$ and $T_2$ such that $T\left(n\right)=T_1\left(n+1\right)-T_1\left(n\right)+T_2\left(n\right)$ and the certificate $\frac{E\left(T_2\right)}{T_2}=\frac{T_2\left(n+1\right)}{T_2\left(n\right)}$ has a rational normal form $\left(z\&comma;r\&comma;s\&comma;u\&comma;v\right)$ with $v$ of minimal degree.

The output from SumDecomposition is a list of two elements $\left[T_1\&comma;T_2\right]$. Both are represented in the form

for some integer $0\le n_0$. The form shown above is called a multiplicative decomposition of the hypergeometric term $T\left(n\right)$.

If the third optional argument k is not specified, the first unused name in the sequence $k,\mathrm{k0},\mathrm{k1},\mathrm{k2},\mathrm{...}$ is used.

If the fourth optional argument newT is specified, it will be assigned an expression in terms of inert Products of the same form as for $T_i$ above that is equivalent to $T$.

If $T$ is a rational function of $n$, then $T_1$ and $T_2$ will be rational functions as well, and the denominator of $T_2$ is of smallest possible degree. In that case, $T_1$ and $T_2$ are not unique, however, and you can use the minimize=v option to impose some additional conditions on $T_1$ and $T_2$ (see below).

Note: If you set infolevel[SumDecomposition] to $3$, Maple prints diagnostics.

$\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right)\&colon;$

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

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

$\mathrm{infolevel}\left[\mathrm{SumDecomposition}\right]≔3\&colon;$

$\mathrm{SumDecomposition}\left(T\&comma;n\&comma;k\&comma;'\mathrm{newT}'\right)$

SumDecomposition:   "calling dterm"
SumDecomposition:   "construct the RCF_1 for the certificate of T"
SumDecomposition:   "construct a regular description of T"
SumDecomposition:   "calling dcert"
SumDecomposition:   "using factorization method"
SumDecomposition:   "construct a regular description of T1"
SumDecomposition:   "construct a regular description of T2"
SumDecomposition:   "T2 is not summable"
SumDecomposition:   "An attempt to control the degree of the numerator"
SumDecomposition:   "construct a triple that regularly describes T2"

$\left[\frac{\left(n+3\right)\left(\prod _{k=1}^{n-1}\frac{2}{k+4}\right)}{12n}\&comma;\frac{\left(n^2+2n-1\right)\left(\prod _{k=1}^{n-1}\frac{2}{k+4}\right)}{12n^2}\right]$

$\mathrm{newT}$

$\frac{\left(n^2-2n-1\right)\left(\prod _{k=1}^{n-1}\frac{2}{k+4}\right)}{12n^2\left(n+1\right)}$

$\mathrm{convert}\left(\mathrm{value}\left(\mathrm{newT}\right)\&comma;'\mathrm{factorial}'\right)$

$\frac{\left(n^2-2n-1\right)2^n}{\left(n+1\right)n^2\left(n+3\right)!}$

$T≔n^3{2}^n$

$T≔n^3{2}^n$

$\mathrm{infolevel}\left[\mathrm{SumDecomposition}\right]≔0\&colon;$

$\mathrm{SumDecomposition}\left(T\&comma;n\&comma;k\right)$

$\left[\frac{\left(2n^3-12n^2+36n-52\right)2^n}{2}\&comma;0\right]$

$T≔-\frac{2\left(925+190n+19n^2+606n^3+72n^5+435n^4+3n^6\right)}{\left(n+5\right)\left(n^2+12n+37\right)\left(n^3+2\right)\left(n-1\right)n}$

$T≔-\frac{2\left(3n^6+72n^5+435n^4+606n^3+19n^2+190n+925\right)}{\left(n+5\right)\left(n^2+12n+37\right)\left(n^3+2\right)\left(n-1\right)n}$

$\mathrm{SumDecomposition}\left(T\&comma;n\right)$

$\left[\frac{10}{n-1}+\frac{5}{n}+\frac{5}{n+1}+\frac{5}{n+2}+\frac{5}{n+3}+\frac{5}{n+4}\&comma;\frac{5}{n-1}+\frac{-4n-22}{n^2+12n+37}+\frac{-n^2-2n-4}{n^3+2}\right]$

$\mathrm{SumDecomposition}\left(T\&comma;n\&comma;'\mathrm{minimize}'=''numerator''\right)$

$\left[\frac{5}{n-1}+\frac{5}{n}+\frac{5}{n+1}+\frac{5}{n+2}+\frac{5}{n+3}+\frac{5}{n+4}+\frac{-4n+2}{{\left(n-6\right)}^2+12n-35}+\frac{-4n-2}{{\left(n-5\right)}^2+12n-23}+\frac{-4n-6}{{\left(n-4\right)}^2+12n-11}+\frac{-4n-10}{{\left(n-3\right)}^2+12n+1}+\frac{-4n-14}{{\left(n-2\right)}^2+12n+13}+\frac{-4n-18}{{\left(n-1\right)}^2+12n+25}\&comma;\frac{5}{n}+\frac{-4n+2}{{\left(n-6\right)}^2+12n-35}+\frac{-n^2-2n-4}{n^3+2}\right]$

$\mathrm{SumDecomposition}\left(T\&comma;n\&comma;'\mathrm{minimize}'=\frac{1}{5}\right)$

$\left[\frac{5}{n-1}-\frac{-n^2-2n-4}{n^3+2}\&comma;\frac{-4n-22}{n^2+12n+37}+\frac{-{\left(n+1\right)}^2-2n-6}{{\left(n+1\right)}^3+2}+\frac{5}{n+5}\right]$

$\mathrm{SumDecomposition}\left(T\&comma;n\&comma;'\mathrm{minimize}'=''sum\; denominator''\right)$

$\left[\frac{5}{n-1}\&comma;\frac{5}{n+5}+\frac{-4n-22}{n^2+12n+37}+\frac{-n^2-2n-4}{n^3+2}\right]$

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

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

$\mathrm{SumDecomposition}\left(T\&comma;n\right)$

$\left[\frac{n-1}{n^2}-\frac{1}{n-1}\&comma;-\frac{n^2+n+3}{n^3}\right]$

$\mathrm{SumDecomposition}\left(T\&comma;n\&comma;'\mathrm{minimize}'=''left''\right)$

$\left[\frac{-2{\left(n-1\right)}^2-n-2}{{\left(n-1\right)}^3}+\frac{n-1}{n^2}\&comma;-\frac{n^2-n+3}{{\left(n-1\right)}^3}\right]$

$\mathrm{SumDecomposition}\left(T\&comma;n\&comma;'\mathrm{minimize}'=''right''\right)$

$\left[-\frac{-2n^2-3}{n^3}-\frac{1}{n-1}\&comma;-\frac{n^2+3n+5}{{\left(n+1\right)}^3}\right]$

Abramov, S.A. "Indefinite Sums of Rational Functions." Proceedings ISSAC'95, pp. 303-308. 1995.

Abramov, S.A., and Petkovsek, M. "Minimal Decomposition of Indefinite Hypergeometric Sums." Proceedings ISSAC'2001, pp. 7-14. 2001.

Abramov, S.A., and Petkovsek, M. "Rational Normal Forms and Minimal Decompositions of Hypergeometric Terms." Journal of Symbolic Computation. Vol. 33 No. 5. (2002): 521-543.

Polyakov, S.P. "Symbolic Additive Decomposition of Rational Functions." Programming and Computer Software, Vol. 31 No. 2. (2005): 60-64.

Polyakov, S.P. "Indefinite Summation of Rational Functions with Additional Minimization of the Summable Part." Programming and Computer Software 34 No. 2, (2008): 95-100.