SumTools[Hypergeometric]
EfficientRepresentation
construct the four efficient representations of a hypergeometric term
Calling Sequence
Parameters
Description
Examples
References
EfficientRepresentation[1](H, n)
EfficientRepresentation[2](H, n)
EfficientRepresentation[3](H, n)
EfficientRepresentation[4](H, n)
H
-
hypergeometric term of n
n
variable
Let H be a hypergeometric term of n. The EfficientRepresentation[i](H,n) calling sequence constructs the ith efficient representation of H of the form Hn=αnVnQn where alpha is a constant, Qn is a product of Gamma-function values and their reciprocals. Additionally,
Qn has the minimal number of factors,
Vn is a rational function which is minimal in one sense or another, depending on the particular rational canonical form chosen to represent the certificate of Hn.
If i=1 then degreedenomV is minimal;
if i=2 then degreenumerV is minimal;
if i=3 then degreenumerV+degreedenomV is minimal, and degreedenomV is minimal;
if i=4 then degreenumerV+degreedenomV is minimal, and degreenumerV is minimal.
If EfficientRepresentation is called without an index, the first efficient representation is constructed.
withSumToolsHypergeometric:
H≔Product123k2+6k+42k+34k+5k+14k+3k4k−12k−14k−32k+5k+23k2+1,k=1..n−1
H≔∏k=1n−13k2+6k+42k+34k+5k+14k+32k4k−12k−14k−32k+5k+23k2+1
EfficientRepresentation1H,n
64π14nn2+13nn−14n+12n+14n−12n−34Γn+52Γn+2
EfficientRepresentation2H,n
64π14nn2+13n−14n+14n−34n+32n+1ΓnΓn−12
EfficientRepresentation3H,n
64π14nn2+13nn−14n+14n−34n+32Γn+2Γn−12
EfficientRepresentation4H,n
RegularGammaFormH,n
64π12nΓn+1−I33Γn+1+I33Γn+32Γn+54Γn+1Γn+342nΓnΓn−14Γn−12Γn−34Γn+52Γn+2Γn−I33Γn+I33
Abramov, S.A.; Le, H.Q.; and Petkovsek, M. "Rational Canonical Forms and Efficient Representations of Hypergeometric Terms." Proc. ISSAC'2003, pp. 7-14. 2003.
See Also
SumTools[Hypergeometric][MultiplicativeDecomposition]
SumTools[Hypergeometric][RationalCanonicalForm]
SumTools[Hypergeometric][RegularGammaForm]
SumTools[Hypergeometric][SumDecomposition]
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