The AccurateSummation(T, n) command solves the problem of indefinite summation using accurate summation.

The output from AccurateSummation is a function $G$ such that $T\left(n\right)=G\left(n+1\right)-G\left(n\right)$ if the algorithm succeeds in constructing one. Otherwise, it returns FAIL.

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

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

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

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

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

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

$\mathrm{FAIL}$

Abramov, S.A. and van Hoeij, M. "Integration of Solutions of Linear Functional Equations." Integral Transformations and Special Functions, (1999): 3-12. Vol. 8. No. 1-2.