SumTools[Hypergeometric]
BottomSequence
bottom sequence of a hypergeometric term
Calling Sequence
Parameters
Description
Examples
References
Compatibility
BottomSequence(T, x, opt)
T
-
hypergeometric term in x
x
name
opt
(optional) equation of the form primitive=true or primitive=false
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 .
Note that is not equivalent to :
Error, numeric exception: division by zero
However, satisfies the same difference equation as :
is an indefinite sum of :
Now assume that :
With that assumption, and are equivalent, and is an indefinite sum of both:
Example of a hypergeometric term with parameters:
Note that is considered non-integer.
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.
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][Gosper]
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