SumTools[Hypergeometric] - Maple Programming Help

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SumTools[Hypergeometric]

 DefiniteSum
 compute the definite sum

 Calling Sequence DefiniteSum(T, n, k, l..u)

Parameters

 T - function of n n - name k - name l..u - range for k

Description

 • For a specified hypergeometric term T of n and k, the DefiniteSum(T, n, k, l..u) command computes, if it exists, a closed form for the definite sum $f\left(n\right)=\sum _{k=l}^{u}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T$.
 • Let r, s, u, v be integers. The DefiniteSum command computes closed forms for four types of definite sums. They are $\sum _{k=nr+s}^{nu+v}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T\left(n,k\right)$, $\sum _{k=nr+s}^{\mathrm{\infty }}T\left(n,k\right)$, $\sum _{k=-\mathrm{\infty }}^{nu+v}T\left(n,k\right)$, and $\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}T\left(n,k\right)$.
 • A closed form is defined as one that can be represented as a sum of hypergeometric terms or as a d'Alembertian term.
 • If the input T is a definite sum of a hypergeometric term, and if the environment variable _EnvDoubleSum is set to true, then DefiniteSum tries to find a closed form for the specified definite sum of T. Note that this operation can be very expensive.
 For more information on the construction of the minimal Z-pair for T, see ExtendedZeilberger.
 Note: If you set infolevel[DefiniteSum] to 3, Maple prints diagnostics.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}[\mathrm{Hypergeometric}]\right):$
 > $T≔\frac{{\left(-1\right)}^{k}\mathrm{binomial}\left(2n,k\right){\mathrm{binomial}\left(2n-k,n\right)}^{2}\left(2n+1\right)}{2n+1+k}:$
 > ${\sum }_{k=0}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T=\mathrm{DefiniteSum}\left(T,n,k,0..n\right)$
 ${\sum }_{{k}{=}{0}}^{{n}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{k}}{}{\mathrm{binomial}}{}\left({2}{}{n}{,}{k}\right){}{{\mathrm{binomial}}{}\left({2}{}{n}{-}{k}{,}{n}\right)}^{{2}}{}\left({2}{}{n}{+}{1}\right)}{{2}{}{n}{+}{1}{+}{k}}{=}\frac{{64}}{{27}}{}\frac{{\mathrm{Γ}}{}\left({2}{}{n}{+}\frac{{3}}{{2}}\right){}{{\mathrm{Γ}}{}\left({n}{+}\frac{{3}}{{2}}\right)}^{{3}}{}{{1024}}^{{n}}{}{{729}}^{{-}{n}}}{{\left({2}{}{n}{+}{1}\right)}^{{2}}{}{{\mathrm{Γ}}{}\left({n}{+}\frac{{4}}{{3}}\right)}^{{2}}{}{{\mathrm{Γ}}{}\left({n}{+}\frac{{2}}{{3}}\right)}^{{2}}{}{\mathrm{Γ}}{}\left({n}{+}{1}\right)}$ (1)
 > $T≔\frac{{\left(-1\right)}^{k}}{\left(k+1\right)\mathrm{binomial}\left(2n,k\right)}:$

Set the infolevel to 3.

 > ${\mathrm{infolevel}}_{\mathrm{DefiniteSum}}≔3:$
 > ${\sum }_{k=0}^{2n-1}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T=\mathrm{DefiniteSum}\left(T,n,k,0..2n-1\right)$
 DefiniteSum:   "try algorithms for definite sum" Definite:   "Construct the Zeilberger's recurrence" Definite:   "Solve the recurrence equation ..." Definite:   "Find hypergeometric solutions" Definite:   "Find a particular d'Alembertian solution" Definite:   "Solve the homogeneous linear recurrence equation" Definite:   "Construction of the general solution successful" Definite:   "Solve the initial-condition problem"
 ${\sum }_{{k}{=}{0}}^{{2}{}{n}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{k}}}{\left({k}{+}{1}\right){}{\mathrm{binomial}}{}\left({2}{}{n}{,}{k}\right)}{=}{-}\frac{{1}}{{8}}{}\left({2}{}{n}{+}{1}\right){}\left({2}{}{\mathrm{Ψ}}{}\left({1}{,}{n}{+}\frac{{1}}{{2}}\right){-}{6}{}{\mathrm{Ψ}}{}\left({1}{,}{n}{+}{1}\right)\right)$ (2)
 > ${\mathrm{infolevel}}_{\mathrm{DefiniteSum}}≔0:$
 > $T≔\frac{{\left(-1\right)}^{k}\mathrm{binomial}\left(n,k\right)\cdot 1}{\mathrm{binomial}\left(x+k,k\right)}:$
 > $T≔{\sum }_{m=0}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\genfrac{}{}{0}{}{T}{\phantom{n=m}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{T}}{n=m}\right)$
 ${T}{≔}{\sum }_{{m}{=}{0}}^{{n}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{k}}{}{\mathrm{binomial}}{}\left({m}{,}{k}\right)}{{\mathrm{binomial}}{}\left({x}{+}{k}{,}{k}\right)}$ (3)
 > $\mathrm{_EnvDoubleSum}≔\mathrm{true}$
 ${\mathrm{_EnvDoubleSum}}{≔}{\mathrm{true}}$ (4)
 > ${\sum }_{k=0}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T=\mathrm{DefiniteSum}\left(T,n,k,0..n\right)$
 ${\sum }_{{k}{=}{0}}^{{n}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{\sum }_{{m}{=}{0}}^{{n}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{k}}{}{\mathrm{binomial}}{}\left({m}{,}{k}\right)}{{\mathrm{binomial}}{}\left({x}{+}{k}{,}{k}\right)}{=}{1}{+}{x}{}{\mathrm{Ψ}}{}\left({x}{+}{n}{+}{1}\right){-}{x}{}{\mathrm{Ψ}}{}\left({x}{+}{1}\right)$ (5)

References

 Abramov, S.A., and Zima, E.V. "D'Alembertian Solutions of Inhomogeneous Linear Equations (differential, difference, and some other)." Proceedings ISSAC'96, pp. 232-240. 1996.
 Petkovsek, M. "Hypergeometric Solutions of Linear Recurrences with Polynomial Coefficients." Journal of Symbolic Computing. Vol. 14. (1992): 243-264.
 van Hoeij, M. "Finite Singularities and Hypergeometric Solutions of Linear Recurrence Equations." Journal of Pure and Applied Algebra. Vol. 139. (1999): 109-131.
 Zeilberger, D. "The Method of Creative Telescoping." Journal of Symbolic Computing. Vol. 11. (1991): 195-204.