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.

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

$T≔\frac{{\left(-1\right)}^k\mathrm{binomial}\left(2n\,k\right){\mathrm{binomial}\left(2n-k\,n\right)}^2\left(2n+1\right)}{2n+1+k}\:$

$\mathrm{Sum}\left(T\,k=0..n\right)=\mathrm{DefiniteSum}\left(T\,n\,k\,0..n\right)$

$\sum _{k=0}^n\frac{{\left(\mathrm{-1}\right)}^k\left(\genfrac{}{}{0ex}{}{2n}{k}\right){\left(\genfrac{}{}{0ex}{}{2n-k}{n}\right)}^2\left(2n+1\right)}{2n+1+k}=\frac{\left(\genfrac{}{}{0ex}{}{2n}{n}\right)\left(\genfrac{}{}{0ex}{}{4n+1}{n}\right)}{\left(\genfrac{}{}{0ex}{}{3n+1}{n}\right)}$

$T≔\frac{{\left(-1\right)}^k}{\left(k+1\right)\mathrm{binomial}\left(2n\,k\right)}\:$

$\mathrm{infolevel}\left[\mathrm{DefiniteSum}\right]≔3\:$

$\mathrm{Sum}\left(T\,k=0..2n-1\right)=\mathrm{DefiniteSum}\left(T\,n\,k\,0..2n-1\right)$

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"

$\sum _{k=0}^{2n-1}\frac{{\left(\mathrm{-1}\right)}^k}{\left(k+1\right)\left(\genfrac{}{}{0ex}{}{2n}{k}\right)}=-\frac{\left(2n+1\right)\left(-3\mathrm{\Psi }\left(1\,n+1\right)+\mathrm{\Psi }\left(1\,n+\frac{1}{2}\right)\right)}{4}$

$\mathrm{infolevel}\left[\mathrm{DefiniteSum}\right]≔0\:$

$T≔\frac{{\left(-1\right)}^k\mathrm{binomial}\left(n\,k\right)\cdot 1}{\mathrm{binomial}\left(x+k\,k\right)}\:$

$T≔\mathrm{Sum}\left(\mathrm{eval}\left(T\,n=m\right)\,m=0..n\right)$

$T≔\sum _{m=0}^n\frac{{\left(\mathrm{-1}\right)}^k\left(\genfrac{}{}{0ex}{}{m}{k}\right)}{\left(\genfrac{}{}{0ex}{}{x+k}{k}\right)}$

$\mathrm{\_EnvDoubleSum}≔\mathrm{true}$

$\mathrm{\_EnvDoubleSum}≔\mathrm{true}$

$\mathrm{Sum}\left(T\,k=0..n\right)=\mathrm{DefiniteSum}\left(T\,n\,k\,0..n\right)$

$\sum _{k=0}^n\sum _{m=0}^n\frac{{\left(\mathrm{-1}\right)}^k\left(\genfrac{}{}{0ex}{}{m}{k}\right)}{\left(\genfrac{}{}{0ex}{}{x+k}{k}\right)}=1+x\mathrm{\Psi }\left(x+n+1\right)-x\mathrm{\Psi }\left(x+1\right)$

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.