construct the summable space
SummableSpace[method](reqn, fcn, options)
SummableSpace[method](cert, n, v, options)
(optional) either Gosper or AccurateSummation; if omitted, Gosper is assumed
homogeneous linear recurrence
function name, e.g., v(n)
rational function in n
name; the independent variable
name; the dependent variable
sequence of optional equations of the form keyword=value. Possible keywords are output, range, or primitive.
Each optional argument is of the form keyword = value. The following options are supported.
Specifies the desired form of representations of sequences in the summable space. Possible values:
Indicates that the sequences are to be represented by an RESol data structure, of the form RESol⁡reqn,v⁡n,inits, where inits is a set of initial conditions.
Indicates that the sequences are to be represented by an explicit expression depending on n, 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.
Specify an interval R=a..b with integer or infinite bounds (−∞..∞ by default). If this option is given then it is assumed that v⁡n is determined only for n∈R and satisfies reqn for all integers n such that both n and n+1 are in R. Moreover, the discrete Newton-Leibniz formula should be valid for any integers n1,n2∈R.
If this option is given, the command returns a pair V,T where V represents the summable space of all v⁡n and T represents the space of all primitives u⁡n. In the Gosper case, both are returned in the form specified by the option 'output'. In the AccurateSummation case, T is returned as an expression in terms of n and v and is typically a piecewise expression. The default is false.
The command SummableSpace(reqn, fcn) or SummableSpace[Gosper](reqn, fcn) constructs the space of all Gosper definite summable sequences v⁡n satisfying the given homogeneous first order linear recurrence reqn with polynomial coefficients, of the form a1⁡n⁢v⁡n+1+a0⁡n⁢v⁡n=0, for all integers n.
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 v⁡0,v⁡1,... 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 denom⁡cert⁢v⁡n+1−numer⁡cert⁢v⁡n=0.
A sequence satisfying a first order linear recurrence is called hypergeometric. A hypergeometric sequence v⁡n is called Gosper indefinite summable if there is another hypergeometric sequence u⁡n such that v⁡n=u⁡n+1−u⁡n. The sequence u⁡n is called a primitive for v⁡n. A Gosper indefinite summable sequence is called Gosper definite summable if the discrete Newton-Leibniz formula
is valid for any integers n1,n2.
A sequence v⁡n satisfying a homogeneous linear recurrence with polynomial coefficients of order d is called accurate summation indefinite summable if there is a sequence u⁡n such that v⁡n=u⁡n+1−u⁡n and u⁡n satisfies another homogeneous linear recurrence if the same order d. The sequence u⁡n is called a primitive for v⁡n. An accurate summation indefinite summable sequence is called accurate summation definite summable if the discrete Newton-Leibniz formula is valid for any integers n1,n2.
The primitive u⁡n is a linear combination of v⁡n,v⁡n+1,...,v⁡n+d−1 with rational function coefficients, where d is the order of reqn, with the possible exception of finitely many values n. In particular, in the Gosper case the primitive is a rational function multiple of v⁡n.
If no nonzero summable sequences for reqn exist, then the command returns FAIL.
rec ≔ k⁢v⁡k+1−k+12⁢v⁡k=0
V,T ≔ SummableSpace'Gosper'⁡rec,v⁡k,'output'='piecewise','primitive'
cert ≔ kk+2
L ≔ k−3⁢k−2⁢k+1⁢v⁡k+2−k−3⁢k2−2⁢k−1⁢v⁡k+1−k−22⁢v⁡k=0
S.A. Abramov. "On the summation of P-recursive sequences." Proc. of ISSAC'06, (2006): 17-22.
The SumTools[DefiniteSum][SummableSpace] command was introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
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