 DefiniteSum - Maple Help

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 $f\left(n\right)=\sum _{k=l}^{u}T$.
 • Let r, s, u, v be integers. The DefiniteSum command computes closed forms for four types of definite sums. They are $\sum _{k=rn+s}^{un+v}T\left(n,k\right)$, $\sum _{k=rn+s}^{\mathrm{\infty }}T\left(n,k\right)$, $\sum _{k=-\mathrm{\infty }}^{un+v}T\left(n,k\right)$, 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

 > $\mathrm{with}\left({\mathrm{SumTools}}_{\mathrm{Hypergeometric}}\right):$
 > $T≔\frac{{\left(-1\right)}^{k}\mathrm{binomial}\left(2n,k\right){\mathrm{binomial}\left(2n-k,n\right)}^{2}\left(2n+1\right)}{2n+1+k}:$
 > ${\sum }_{k=0}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T=\mathrm{DefiniteSum}\left(T,n,k,0..n\right)$
 ${\sum }_{{k}{=}{0}}^{{n}}{}\frac{{\left({-1}\right)}^{{k}}{}\left(\genfrac{}{}{0}{}{{2}{}{n}}{{k}}\right){}{\left(\genfrac{}{}{0}{}{{2}{}{n}{-}{k}}{{n}}\right)}^{{2}}{}\left({2}{}{n}{+}{1}\right)}{{2}{}{n}{+}{1}{+}{k}}{=}\frac{\left(\genfrac{}{}{0}{}{{2}{}{n}}{{n}}\right){}\left(\genfrac{}{}{0}{}{{4}{}{n}{+}{1}}{{n}}\right)}{\left(\genfrac{}{}{0}{}{{3}{}{n}{+}{1}}{{n}}\right)}$ (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 recurrence" Definite:   "Solve the recurrence equation ..." Definite:   "Find hypergeometric solutions" Definite:   "Solve the homogeneous linear recurrence equation" Definite:   "Find a particular hypergeometric solution" Definite:   "Find a particular d'Alembertian solution" Definite:   "Construction of the general solution successful" Definite:   "Solve the initial-condition problem"
 ${\sum }_{{k}{=}{0}}^{{2}{}{n}{-}{1}}{}\frac{{\left({-1}\right)}^{{k}}}{\left({k}{+}{1}\right){}\left(\genfrac{}{}{0}{}{{2}{}{n}}{{k}}\right)}{=}{-}\frac{\left({2}{}{n}{+}{1}\right){}\left({-}{3}{}{\mathrm{\Psi }}{}\left({1}{,}{n}{+}{1}\right){+}{\mathrm{\Psi }}{}\left({1}{,}{n}{+}\frac{{1}}{{2}}\right)\right)}{{4}}$ (2)
 > ${\mathrm{infolevel}}_{\mathrm{DefiniteSum}}≔0:$
 > $T≔\frac{{\left(-1\right)}^{k}\mathrm{binomial}\left(n,k\right)\cdot 1}{\mathrm{binomial}\left(x+k,k\right)}:$
 > $T≔{\sum }_{m=0}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\genfrac{}{}{0}{}{T}{\phantom{n=m}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{T}}{n=m}\right)$
 ${T}{≔}{\sum }_{{m}{=}{0}}^{{n}}{}\frac{{\left({-1}\right)}^{{k}}{}\left(\genfrac{}{}{0}{}{{m}}{{k}}\right)}{\left(\genfrac{}{}{0}{}{{x}{+}{k}}{{k}}\right)}$ (3)
 > $\mathrm{_EnvDoubleSum}≔\mathrm{true}$
 ${\mathrm{_EnvDoubleSum}}{≔}{\mathrm{true}}$ (4)
 > ${\sum }_{k=0}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T=\mathrm{DefiniteSum}\left(T,n,k,0..n\right)$
 ${\sum }_{{k}{=}{0}}^{{n}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{m}{=}{0}}^{{n}}{}\frac{{\left({-1}\right)}^{{k}}{}\left(\genfrac{}{}{0}{}{{m}}{{k}}\right)}{\left(\genfrac{}{}{0}{}{{x}{+}{k}}{{k}}\right)}{=}{1}{+}{x}{}{\mathrm{\Psi }}{}\left({x}{+}{n}{+}{1}\right){-}{x}{}{\mathrm{\Psi }}{}\left({x}{+}{1}\right)$ (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.