SumTools[DefiniteSum][SummableSpace] - construct the summable space
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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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Description
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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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is valid for any integers .
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If no nonzero summable sequences for reqn exist, then the command returns .
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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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Compatibility
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The SumTools[DefiniteSum][SummableSpace] command was introduced in Maple 15.
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Examples
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![RESol({(-k^2+4*k-4)*v(k)+(-k^3+5*k^2-5*k-3)*v(k+1)+(k^3-4*k^2+k+6)*v(k+2) = 0}, {v(k)}, {v(2) = v(2), v(3) = 0, v(4) = v(4), v(5) = -(1/4)*v(4)}, INFO), PIECEWISE([v(k)/(k-3)+k*v(k+1), k <= 2], [0, k = 3], [v(k)/(k-3)+k*v(k+1), 4 <= k])](/support/helpjp/helpview.aspx?si=5468/file06610/math442.png)
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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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