The Rational(f, k) command computes a closed form of the indefinite sum of $f$ with respect to $k$.

Rational functions are summed using Abramov's algorithm (see the References section). For the input rational function $f\left(k\right)$, the algorithm computes two rational functions $s\left(k\right)$ and $t\left(k\right)$ such that $f\left(k\right)=s\left(k+1\right)-s\left(k\right)+t\left(k\right)$ and the denominator of $t\left(k\right)$ has minimal degree with respect to $k$.  The non-rational part, $\sum _{k}t\left(k\right)$, is then expressed in terms of the digamma and polygamma functions.

If the option failpoints=true (or just failpoints for short) is specified, then the command returns a pair $g,\left[p\,q\right]$, where

$g$ is the closed form of the indefinite sum of $f$ w.r.t. $k$,

$p$ is a list containing the integer poles of $f$, and

$q$ is a list containing the poles of $s$ and $t$ that are not poles of $f$.

See SumTools[IndefiniteSum][Indefinite] for more detailed help.

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

$f≔\frac{1}{n^2+\mathrm{sqrt}\left(5\right)n-1}$

$f≔\frac{1}{n^2+\sqrt{5}n-1}$

$g≔\mathrm{Rational}\left(f\,n\right)$

$g≔-\frac{1}{3\left(n-\frac{3}{2}+\frac{\sqrt{5}}{2}\right)}-\frac{1}{3\left(n-\frac{1}{2}+\frac{\sqrt{5}}{2}\right)}-\frac{1}{3\left(n+\frac{1}{2}+\frac{\sqrt{5}}{2}\right)}$

$\mathrm{evala}\left(\mathrm{Normal}\left(\mathrm{eval}\left(g\,n=n+1\right)-g\right)\,\mathrm{expanded}\right)$

$\frac{1}{n^2+\sqrt{5}n-1}$

$f≔\frac{13-57x+2y+20x^2-18xy+10y^2}{15+10x-26y-25x^2+10xy+8y^2}$

$f≔\frac{20x^2-18xy+10y^2-57x+2y+13}{-25x^2+10xy+8y^2+10x-26y+15}$

$g≔\mathrm{Rational}\left(f\,x\right)$

$g≔-\frac{4x}{5}+\left(-\frac{7y}{25}+\frac{34}{25}\right)\mathrm{\Psi }\left(x-\frac{4y}{5}+\frac{3}{5}\right)+\left(\frac{17y}{25}+\frac{3}{5}\right)\mathrm{\Psi }\left(x+\frac{2y}{5}-1\right)$

$\mathrm{simplify}\left(\mathrm{combine}\left(f-\left(\mathrm{eval}\left(g\,x=x+1\right)-g\right)\,\mathrm{\Psi }\right)\right)$

$0$

$f≔\frac{1}{n}-\frac{2}{n-3}+\frac{1}{n-5}$

$f≔\frac{1}{n}-\frac{2}{n-3}+\frac{1}{n-5}$

$g,\mathrm{fp}≔\mathrm{Rational}\left(f\,n\,'\mathrm{failpoints}'\right)$

$g,\mathrm{fp}≔-\frac{1}{n-5}-\frac{1}{n-4}+\frac{1}{n-3}+\frac{1}{n-2}+\frac{1}{n-1},\left[\left[0..0\,3..3\,5..5\right]\,\left[1\,2\,4\right]\right]$

Abramov, S.A. "Indefinite sums of rational functions." Proceedings ISSAC'95, pp. 303-308. 1995.