AccurateSummation - Maple Help

SumTools[IndefiniteSum]

 AccurateSummation
 compute closed forms of indefinite sums using accurate summation

 Calling Sequence AccurateSummation(T, n)

Parameters

 T - function of n n - name; specifies summation index

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{IndefiniteSum}\right]\right):$
 > $T≔\mathrm{\Gamma }\left(n+1\right)-\mathrm{\Gamma }\left(n\right)-\mathrm{\Psi }\left(n\right)$
 ${T}{≔}{\mathrm{\Gamma }}{}\left({n}{+}{1}\right){-}{\mathrm{\Gamma }}{}\left({n}\right){-}{\mathrm{\Psi }}{}\left({n}\right)$ (1)
 > $\mathrm{AccurateSummation}\left(T,n\right)$
 $\frac{\left({{n}}^{{4}}{-}{{n}}^{{3}}{-}{6}{}{{n}}^{{2}}{-}{6}{}{n}{-}{5}\right){}\left({\mathrm{\Gamma }}{}\left({n}{+}{1}\right){-}{\mathrm{\Gamma }}{}\left({n}\right){-}{\mathrm{\Psi }}{}\left({n}\right)\right)}{{{n}}^{{2}}{+}{n}{+}{3}}{-}\frac{\left({{n}}^{{5}}{-}{{n}}^{{4}}{-}{10}{}{{n}}^{{3}}{-}{9}{}{{n}}^{{2}}{-}{2}\right){}\left({\mathrm{\Gamma }}{}\left({n}{+}{2}\right){-}{\mathrm{\Gamma }}{}\left({n}{+}{1}\right){-}{\mathrm{\Psi }}{}\left({n}{+}{1}\right)\right)}{{n}{}\left({{n}}^{{2}}{+}{n}{+}{3}\right)}{+}\frac{\left({n}{+}{1}\right){}\left({{n}}^{{3}}{-}{5}{}{{n}}^{{2}}{+}{4}{}{n}{-}{2}\right){}\left({\mathrm{\Gamma }}{}\left({n}{+}{3}\right){-}{\mathrm{\Gamma }}{}\left({n}{+}{2}\right){-}{\mathrm{\Psi }}{}\left({n}{+}{2}\right)\right)}{{n}{}\left({{n}}^{{2}}{+}{n}{+}{3}\right)}$ (2)

Note that since T is not a hypergeometric term of n, Gosper's algorithm fails to compute an anti-difference:

 > $\mathrm{Hypergeometric}\left(T,n\right)$
 ${\mathrm{FAIL}}$ (3)

References

 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.