SumTools[Hypergeometric] - Maple Programming Help

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SumTools[Hypergeometric]

 IndefiniteSum
 calculate indefinite sum

 Calling Sequence IndefiniteSum(T, n)

Parameters

 T - function of n n - variable

Description

 • The IndefiniteSum(T,n) command computes a function G such that $T\left(n\right)=\left({E}_{n}-1\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}G\left(n\right)$ 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.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}[\mathrm{Hypergeometric}]\right):$
 > $T≔\frac{1}{{n}^{2}+\sqrt{5}n-1}$
 ${T}{≔}\frac{{1}}{{{n}}^{{2}}{+}\sqrt{{5}}{}{n}{-}{1}}$ (1)
 > ${\sum }_{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T=\mathrm{IndefiniteSum}\left(T,n\right)$
 ${\sum }_{{n}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{1}}{{{n}}^{{2}}{+}\sqrt{{5}}{}{n}{-}{1}}{=}{-}\frac{{1}}{{3}{}\left({n}{-}\frac{{3}}{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{5}}\right)}{-}\frac{{1}}{{3}{}\left({n}{-}\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{5}}\right)}{-}\frac{{1}}{{3}{}\left({n}{+}\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{5}}\right)}$ (2)
 > $T≔{n}^{3}{2}^{n}$
 ${T}{≔}{{n}}^{{3}}{}{{2}}^{{n}}$ (3)
 > ${\sum }_{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T=\mathrm{IndefiniteSum}\left(T,n\right)$
 ${\sum }_{{n}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{{n}}^{{3}}{}{{2}}^{{n}}{=}\left({{n}}^{{3}}{-}{6}{}{{n}}^{{2}}{+}{18}{}{n}{-}{26}\right){}{{2}}^{{n}}$ (4)
 > $T≔\mathrm{Γ}\left(n+1\right)-\mathrm{Γ}\left(n\right)-\mathrm{Ψ}\left(n\right)$
 ${T}{≔}{\mathrm{Γ}}{}\left({n}{+}{1}\right){-}{\mathrm{Γ}}{}\left({n}\right){-}{\mathrm{Ψ}}{}\left({n}\right)$ (5)
 > ${\sum }_{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T=\mathrm{IndefiniteSum}\left(T,n\right)$
 ${\sum }_{{n}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\mathrm{Γ}}{}\left({n}{+}{1}\right){-}{\mathrm{Γ}}{}\left({n}\right){-}{\mathrm{Ψ}}{}\left({n}\right)\right){=}\frac{\left({{n}}^{{4}}{-}{{n}}^{{3}}{-}{6}{}{{n}}^{{2}}{-}{6}{}{n}{-}{5}\right){}\left({\mathrm{Γ}}{}\left({n}{+}{1}\right){-}{\mathrm{Γ}}{}\left({n}\right){-}{\mathrm{Ψ}}{}\left({n}\right)\right)}{{{n}}^{{2}}{+}{n}{+}{3}}{-}\frac{\left({{n}}^{{5}}{-}{{n}}^{{4}}{-}{10}{}{{n}}^{{3}}{-}{9}{}{{n}}^{{2}}{-}{2}\right){}\left({\mathrm{Γ}}{}\left({n}{+}{2}\right){-}{\mathrm{Γ}}{}\left({n}{+}{1}\right){-}{\mathrm{Ψ}}{}\left({n}{+}{1}\right)\right)}{{n}{}\left({{n}}^{{2}}{+}{n}{+}{3}\right)}{+}\frac{\left({n}{+}{1}\right){}\left({{n}}^{{3}}{-}{5}{}{{n}}^{{2}}{+}{4}{}{n}{-}{2}\right){}\left({\mathrm{Γ}}{}\left({n}{+}{3}\right){-}{\mathrm{Γ}}{}\left({n}{+}{2}\right){-}{\mathrm{Ψ}}{}\left({n}{+}{2}\right)\right)}{{n}{}\left({{n}}^{{2}}{+}{n}{+}{3}\right)}$ (6)

References

 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.