bottom sequence of a hypergeometric term
BottomSequence(T, x, opt)
hypergeometric term in x
(optional) equation of the form primitive=true or primitive=false
Consider T as an analytic function in x satisfying a linear difference equation p⁡x⁢T⁡x+1+q⁡x⁢T⁡x=0, where p⁡x and q⁡x are polynomials in x. For h∈ℤ and any integer k, let ck,h be the h-th coefficient of the Laurent series expansion for T at x=k. An integer m is called depth of T if ck,h=0 for all h<m and all integers k, and ck,m≠0 for some k∈ℤ.
The bottom sequence of T is the doubly infinite sequence bx defined as bx=cx,m for all integers x, where m is the depth of T. The command BottomSequence(T, x) returns the bottom sequence of T in form of an expression representing a function of (integer values of) x. Typically, this is a piecewise expression.
The bottom sequence bx is defined at all integers x and satisfies the same difference equation p⁡x⁢bx+1+q⁡x⁢bx=0 as T.
If T is Gosper-summable and S=v⁢T is its indefinite sum found by Gosper's algorithm, then the depth of S is also m. If the optional argument primitive=true (or just primitive) is specified, the command returns a pair v,u, where v is the bottom sequence of T and u is the bottom sequence of S or FAIL if T is not Gosper-summable.
Note that this command rewrites expressions of the form nk in terms of GAMMA functions Γ⁡n+1Γ⁡k+1⁢Γ⁡n−k+1.
If assumptions of the form x0<x and/or x<x1 are made, the depth and the bottom of T are computed with respect to the given interval instead of −∞..∞.
T ≔ n⁢n!
b,s ≔ BottomSequence⁡T,n,'primitive'
Note that b is not equivalent to T:
Error, numeric exception: division by zero
However, b satisfies the same difference equation as T:
z ≔ n⁢bn=n+1|bn=n+1−n+12⁢b
s is an indefinite sum of b:
z ≔ sn=n+1|sn=n+1−s−b
Now assume that 0≤n:
b,s ≔ BottomSequence⁡T,n,'primitive'assuming0≤n
With that assumption, b and T are equivalent, and s is an indefinite sum of both:
Example of a hypergeometric term with parameters:
T ≔ GAMMA⁡−nn−k
Note that k is considered non-integer.
Warning, the assumptions about variable(s) k are ignored
T ≔ binomial⁡2⁢n−3,n4n
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.
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