SumTools[Hypergeometric] - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Discrete Mathematics : Summation and Difference Equations : SumTools : SumTools/Hypergeometric/EfficientRepresentation

SumTools[Hypergeometric]

  

EfficientRepresentation

  

construct the four efficient representations of a hypergeometric term

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

EfficientRepresentation[1](H, n)

EfficientRepresentation[2](H, n)

EfficientRepresentation[3](H, n)

EfficientRepresentation[4](H, n)

Parameters

H

-

hypergeometric term of n

n

-

variable

Description

• 

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,

1. 

Qn has the minimal number of factors,

2. 

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.

Examples

withSumTools[Hypergeometric]:

Hk=1n113k2+6k+42k+34k+5k+14k+32k4k12k14k32k+5k+23k2+1

H:=k=1n1123k2+6k+42k+34k+5k+14k+3k4k12k14k32k+5k+23k2+1

(1)

EfficientRepresentation[1]H,n

64π14nn2+13nn14n+12n+14n12n34Γn+52Γn+2

(2)

EfficientRepresentation[2]H,n

64π14nn2+13n14n+14n34n+32n+1ΓnΓn12

(3)

EfficientRepresentation[3]H,n

64π14nn2+13nn14n+14n34n+32Γn+2Γn12

(4)

EfficientRepresentation[4]H,n

64π14nn2+13n14n+14n34n+32n+1ΓnΓn12

(5)

RegularGammaFormH,n

64π12nΓn+113I3Γn+1+13I3Γn+32Γn+54Γn+1Γn+342nΓnΓn14Γn12Γn34Γn+52Γn+2Γn13I3Γn+13I3

(6)

References

  

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]

SumTools[Hypergeometric][MultiplicativeDecomposition]

SumTools[Hypergeometric][RationalCanonicalForm]

SumTools[Hypergeometric][RegularGammaForm]

SumTools[Hypergeometric][SumDecomposition]

 


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam