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Calling Sequence
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SummableSpace[method](reqn, fcn, options)
SummableSpace[method](cert, n, v, options)
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Parameters
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method
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(optional) either Gosper or AccurateSummation; if omitted, Gosper is assumed
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reqn
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homogeneous linear recurrence
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fcn
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function name, e.g., v(n)
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cert
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rational function in n
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n
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name; the independent variable
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v
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name; the dependent variable
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opts
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sequence of optional equations of the form keyword=value. Possible keywords are output, range, or primitive.
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Options
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Each optional argument is of the form keyword = value. The following options are supported.
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Specifies the desired form of representations of sequences in the summable space. Possible values:
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Indicates that the sequences are to be represented by an RESol data structure, of the form , where inits is a set of initial conditions.
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Indicates that the sequences are to be represented by an explicit expression depending on , which in general is a piecewise expression.
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This argument is ignored in the AccurateSummation case, and an RESol data structure is returned always. In the Gosper case, the default is piecewise.
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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 .
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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.
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Description
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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 .
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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.
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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.
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Instead of the recurrence, a certificate cert can be specified, in which case the recurrence is taken as .
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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
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is valid for any integers .
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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 .
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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 .
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If no nonzero summable sequences for reqn exist, then the command returns .
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Examples
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References
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S.A. Abramov. "On the summation of P-recursive sequences." Proc. of ISSAC'06, (2006): 17-22.
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Compatibility
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The SumTools[DefiniteSum][SummableSpace] command was introduced in Maple 15.
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