The IndefiniteSum(T,n) command computes a function G such that $T\left(n\right)=\left(E_n-1\right)·G\left(n\right)$ if it exists.

The classes of functions T supported are rational functions, hypergeometric terms, and those for which the minimal annihilator in ${K\left(n\right)}_{E_n}$ for T can be computed.

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

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

$T≔\frac{1}{n^2+\sqrt{5}n-1}$

$\mathrm{Sum}\left(T\,n\right)=\mathrm{IndefiniteSum}\left(T\,n\right)$

$\sum _n\frac{1}{n^2+\sqrt{5}n-1}=-\frac{1}{3\left(n-\frac{3}{2}+\frac{\sqrt{5}}{2}\right)}-\frac{1}{3\left(n-\frac{1}{2}+\frac{\sqrt{5}}{2}\right)}-\frac{1}{3\left(n+\frac{1}{2}+\frac{\sqrt{5}}{2}\right)}$

$T≔n^3{2}^n$

$T≔n^3{2}^n$

$\mathrm{Sum}\left(T\,n\right)=\mathrm{IndefiniteSum}\left(T\,n\right)$

$\sum _n{n}^3{2}^n=\left(n^3-6n^2+18n-26\right)2^n$

$T≔\mathrm{\Gamma }\left(n+1\right)-\mathrm{\Gamma }\left(n\right)-\mathrm{\Psi }\left(n\right)$

$T≔\mathrm{\Gamma }\left(n+1\right)-\mathrm{\Gamma }\left(n\right)-\mathrm{\Psi }\left(n\right)$

$\mathrm{Sum}\left(T\,n\right)=\mathrm{IndefiniteSum}\left(T\,n\right)$

$\sum _n\left(\mathrm{\Gamma }\left(n+1\right)-\mathrm{\Gamma }\left(n\right)-\mathrm{\Psi }\left(n\right)\right)=\frac{\left(n^4-n^3-6n^2-6n-5\right)\left(\mathrm{\Gamma }\left(n+1\right)-\mathrm{\Gamma }\left(n\right)-\mathrm{\Psi }\left(n\right)\right)}{n^2+n+3}-\frac{\left(n^5-n^4-10n^3-9n^2-2\right)\left(\mathrm{\Gamma }\left(n+2\right)-\mathrm{\Gamma }\left(n+1\right)-\mathrm{\Psi }\left(n+1\right)\right)}{n\left(n^2+n+3\right)}+\frac{\left(n+1\right)\left(n^3-5n^2+4n-2\right)\left(\mathrm{\Gamma }\left(n+3\right)-\mathrm{\Gamma }\left(n+2\right)-\mathrm{\Psi }\left(n+2\right)\right)}{n\left(n^2+n+3\right)}$

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

Abramov, S.A., and van Hoeij, M. "Integration of solutions of linear functional equations." Integral Transformations and Special Functions, Vol. 8 No. 1-2, (1999): 3-12.

Gosper, R.W., Jr. "Decision procedure for indefinite hypergeometric summation." Proc. Natl. Acad. Sci. USA, Vol. 75, (1977): 40-42.