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 .

Let r, s, u, v be integers. The DefiniteSum command computes closed forms for four types of definite sums. They are , , , and .

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.

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"

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.