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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 RESolreqn,vn,inits, where inits is a set of initial conditions.

– 

'piecewise'

  

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.

• 

'range'=a..b

  

Specify an interval R=a..b with integer or infinite bounds (.. by default). If this option is given then it is assumed that vn is determined only for nR 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,n2R.

• 

'primitive'=truefalse

  

If this option is given, the command returns a pair V,T where V represents the summable space of all vn and T represents the space of all primitives un. 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.

Description

• 

The command SummableSpace(reqn, fcn) or SummableSpace[Gosper](reqn, fcn) constructs the space of all Gosper definite summable sequences vn satisfying the given homogeneous first order linear recurrence reqn with polynomial coefficients, of the form a1nvn+1+a0nvn=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 v0,v1,... 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 denomcertvn+1numercertvn=0.

• 

A sequence satisfying a first order linear recurrence is called hypergeometric. A hypergeometric sequence vn is called Gosper indefinite summable if there is another hypergeometric sequence un such that vn=un+1un. The sequence un is called a primitive for vn. A Gosper indefinite summable sequence is called Gosper definite summable if the discrete Newton-Leibniz formula

n=n1n2vn=un2+1un1

  

is valid for any integers n1,n2.

• 

A sequence vn satisfying a homogeneous linear recurrence with polynomial coefficients of order d is called accurate summation indefinite summable if there is a sequence un such that vn=un+1un and un satisfies another homogeneous linear recurrence if the same order d. The sequence un is called a primitive for vn. 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 un is a linear combination of vn,vn+1,...,vn+d1 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 vn.

• 

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

Examples

withSumToolsDefiniteSum:

reckvk+1k+12vk=0

reckvk+1k+12vk=0

(1)

SummableSpacerec,vk,output=RESol

RESolk22k1vk+kvk+1=0,vk,v−1=v−1,v0=0,v1=0,INFO

(2)

V,TSummableSpaceGosperrec,vk,output=piecewise,primitive

V,Tv−1−1kkΓkk−100k,v−1−1kΓkk−100k

(3)

addevalV,k=i,i=100..100=evalT,k=101evalT,k=100

v−1933262154439441526816992388562667004907159682643816214685929638952175999932299156089414639761565182862536979208272237582511852109168640000000000000000000000=v−1933262154439441526816992388562667004907159682643816214685929638952175999932299156089414639761565182862536979208272237582511852109168640000000000000000000000

(4)

SummableSpaceGosperrec,vk,range=0..

v1Γk+1k

(5)

certkk+2

certkk+2

(6)

SummableSpaceGospercert,k,v

0k−2v−1k=−1v−1k=001k

(7)

SummableSpaceGospercert,k,v,range=1..

2v1k+1k

(8)

SummableSpaceGosper2k24k9vk+12k3k1k8vk=0,vk

0k−22v1k=−13v1k=0v1k=18v3Γk32k2k9πΓk+22k

(9)

Lk3k2k+1vk+2k3k22k1vk+1k22vk=0

Lk3k2k+1vk+2k3k22k1vk+1k22vk=0

(10)

SummableSpaceAccurateSummationL,vk,primitive

RESolk2+4k4vk+k3+5k25k3vk+1+k34k2+k+6vk+2=0,vk,v2=v2,v3=0,v4=v4,v5=v44,INFO,vkk3+kvk+1k20k=3vkk3+kvk+14k

(11)

SummableSpaceAccurateSummationL,vk,range=4..,primitive

RESolk2+4k4vk+k3+5k25k3vk+1+k34k2+k+6vk+2=0,vk,v4=v4,v5=v5,INFO,vkk3+kvk+1

(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]