SummableSpace - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

SumTools[DefiniteSum]

  

SummableSpace

  

construct the summable space

 

Calling Sequence

Parameters

Options

Description

Examples

References

Compatibility

Calling Sequence

SummableSpace[method](reqn, fcn, options)

SummableSpace[method](cert, n, v, options)

Parameters

method

-

(optional) either Gosper or AccurateSummation; if omitted, Gosper is assumed

reqn

-

homogeneous linear recurrence

fcn

-

function name, e.g., v(n)

cert

-

rational function in n

n

-

name; the independent variable

v

-

name; the dependent variable

opts

-

sequence of optional equations of the form keyword=value. Possible keywords are output, range, or primitive.

Options

• 

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.

Description

• 

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 .

Examples

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

References

  

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

Compatibility

• 

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

• 

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

See Also

OreTools[MathOperations][AccurateIntegration]

SumTools[Hypergeometric][BottomSequence]

SumTools[Hypergeometric][Gosper]

SumTools[IndefiniteSum][AccurateSummation]

 


Download Help Document