BottomSequence - Maple Help
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SumTools[Hypergeometric]

  

BottomSequence

  

bottom sequence of a hypergeometric term

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

BottomSequence(T, x, opt)

Parameters

T

-

hypergeometric term in x

x

-

name

opt

-

(optional) equation of the form primitive=true or primitive=false

Description

• 

Consider  as an analytic function in  satisfying a linear difference equation , where  and  are polynomials in . For  and any integer , let  be the -th coefficient of the Laurent series expansion for  at . An integer  is called depth of  if  for all  and all integers , and  for some .

• 

The bottom sequence of  is the doubly infinite sequence  defined as  for all integers , where  is the depth of . The command BottomSequence(T, x) returns the bottom sequence of  in form of an expression representing a function of (integer values of) . Typically, this is a piecewise expression.

• 

The bottom sequence  is defined at all integers  and satisfies the same difference equation  as .

• 

If  is Gosper-summable and  is its indefinite sum found by Gosper's algorithm, then the depth of  is also . If the optional argument primitive=true (or just primitive) is specified, the command returns a pair , where  is the bottom sequence of  and  is the bottom sequence of  or FAIL if  is not Gosper-summable.

• 

Note that this command rewrites expressions of the form  in terms of GAMMA functions .

• 

If assumptions of the form  and/or  are made, the depth and the bottom of  are computed with respect to the given interval instead of .

Examples

(1)

(2)

Note that  is not equivalent to :

(3)

(4)

(5)

Error, numeric exception: division by zero

However,  satisfies the same difference equation as :

(6)

(7)

(8)

(9)

(10)

 is an indefinite sum of :

(11)

(12)

(13)

(14)

Now assume that :

(15)

With that assumption,  and  are equivalent, and  is an indefinite sum of both:

(16)

(17)

Example of a hypergeometric term with parameters:

(18)

(19)

Note that  is considered non-integer.

(20)

(21)

(22)

(23)

References

  

S.A. Abramov, M. Petkovsek. "Analytic solutions of linear difference equations, formal series, and bottom summation." Proc. of CASC'07, (2007): 1-10.

  

S.A. Abramov, M. Petkovsek. "Gosper's Algorithm, Accurate Summation, and the Discrete Newton-Leibniz Formula." Proceedings of ISSAC'05, (2005): 5-12.

Compatibility

• 

The SumTools[Hypergeometric][BottomSequence] command was introduced in Maple 15.

• 

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

See Also

assuming

binomial

SumTools[DefiniteSum][SummableSpace]

SumTools[Hypergeometric]

SumTools[Hypergeometric][Gosper]

 


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