Consider  as an analytic function in  satisfying a linear difference equation , where  and  are polynomials in . For  and any integer , let  be the -th coefficient of the Laurent series expansion for  at . An integer  is called depth of  if  for all  and all integers , and  for some .

The bottom sequence of  is the doubly infinite sequence  defined as  for all integers , where  is the depth of . The command BottomSequence(T, x) returns the bottom sequence of  in form of an expression representing a function of (integer values of) . Typically, this is a piecewise expression.

The bottom sequence  is defined at all integers  and satisfies the same difference equation  as .

If  is Gosper-summable and  is its indefinite sum found by Gosper's algorithm, then the depth of  is also . If the optional argument primitive=true (or just primitive) is specified, the command returns a pair , where  is the bottom sequence of  and  is the bottom sequence of  or FAIL if  is not Gosper-summable.

Note that this command rewrites expressions of the form  in terms of GAMMA functions .

If assumptions of the form  and/or  are made, the depth and the bottom of  are computed with respect to the given interval instead of .

The SumTools[Hypergeometric][BottomSequence] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.

Error, numeric exception: division by zero

Warning, the assumptions about variable(s) k are ignored

S.A. Abramov, M. Petkovsek. "Analytic solutions of linear difference equations, formal series, and bottom summation." Proc. of CASC'07, (2007): 1-10.

S.A. Abramov, M. Petkovsek. "Gosper's Algorithm, Accurate Summation, and the Discrete Newton-Leibniz Formula." Proceedings of ISSAC'05, (2005): 5-12.