The command SummableSpace(reqn, fcn) or SummableSpace[Gosper](reqn, fcn) constructs the space of all Gosper definite summable sequences  satisfying the given homogeneous first order linear recurrence reqn with polynomial coefficients, of the form , for all integers .

The command SummableSpace[AccurateSummation](reqn, fcn) constructs the space of accurate summation definite summable sequences satisfying a given homogeneous linear recurrence reqn of arbitrary order with polynomial coefficients.

The form in which the result is returned is determined by the output option; see below for details. The output may contain placeholders of the form  representing initial conditions or free parameters of the resulting space.

Instead of the recurrence, a certificate cert can be specified, in which case the recurrence is taken as .

A sequence satisfying a first order linear recurrence is called hypergeometric. A hypergeometric sequence  is called Gosper indefinite summable if there is another hypergeometric sequence  such that . The sequence  is called a primitive for . A Gosper indefinite summable sequence is called Gosper definite summable if the discrete Newton-Leibniz formula

is valid for any integers .

A sequence  satisfying a homogeneous linear recurrence with polynomial coefficients of order  is called accurate summation indefinite summable if there is a sequence  such that  and  satisfies another homogeneous linear recurrence if the same order . The sequence  is called a primitive for . An accurate summation indefinite summable sequence is called accurate summation definite summable if the discrete Newton-Leibniz formula is valid for any integers .

The primitive  is a linear combination of  with rational function coefficients, where  is the order of reqn, with the possible exception of finitely many values . In particular, in the Gosper case the primitive is a rational function multiple of .

If no nonzero summable sequences for reqn exist, then the command returns .

Each optional argument is of the form keyword = value. The following options are supported.

'output'

Specifies the desired form of representations of sequences in the summable space. Possible values:

'RESol'

Indicates that the sequences are to be represented by an RESol data structure, of the form , where inits is a set of initial conditions.

'piecewise'

Indicates that the sequences are to be represented by an explicit expression depending on , which in general is a piecewise expression.

This argument is ignored in the AccurateSummation case, and an RESol data structure is returned always. In the Gosper case, the default is piecewise.

'range'=a..b

Specify an interval  with integer or infinite bounds ( by default). If this option is given then it is assumed that  is determined only for  and satisfies reqn for all integers  such that both  and  are in . Moreover, the discrete Newton-Leibniz formula should be valid for any integers .

'primitive'=truefalse

If this option is given, the command returns a pair  where  represents the summable space of all  and  represents the space of all primitives . In the Gosper case, both are returned in the form specified by the option 'output'. In the AccurateSummation case,  is returned as an expression in terms of  and  and is typically a piecewise expression. The default is false.

The SumTools[DefiniteSum][SummableSpace] command was introduced in Maple 15.

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

S.A. Abramov. "On the summation of P-recursive sequences." Proc. of ISSAC'06, (2006): 17-22.