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SumTools[Hypergeometric]

  

RegularGammaForm

  

construct the regular Gamma-function representation of a hypergeometric term

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

RegularGammaForm(H, n)

Parameters

H

-

hypergeometric term of n

n

-

variable

Description

• 

Let H be a hypergeometric term of n, R be the certificate of H, and n0 be an integer such that R has neither a pole nor a zero for all n0n. The RegularGammaForm(H,n) calling sequence returns the multiplicative decomposition of the form Hn0k=n0n1Rk where the product is expressed in terms of a product of the Gamma function of the form Γnc where c is a constant and their reciprocals.

Examples

withSumToolsHypergeometric:

HProduct123k2+6k+42k+34k+5k+14k+3k4k12k14k32k+5k+23k2+1,k=1..n1

Hk=1n13k2+6k+42k+34k+5k+14k+32k4k12k14k32k+5k+23k2+1

(1)

RegularGammaFormH,n

64π12nΓn+1I33Γn+1+I33Γn+32Γn+54Γn+1Γn+342nΓnΓn14Γn12Γn34Γn+52Γn+2ΓnI33Γn+I33

(2)

Compare the number of Gamma-function values returned from RegularGammaForm with that of any one of the four efficient representations of the input hypergeometric term Hn:

EfficientRepresentation1H,n

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

(3)

EfficientRepresentation2H,n

64π14nn2+13n14n+14n34n+32n+1ΓnΓn12

(4)

EfficientRepresentation3H,n

64π14nn14n2+13nn+14n34n+32Γn12Γn+2

(5)

EfficientRepresentation4H,n

64π14nn2+13n14n+14n34n+32n+1ΓnΓn12

(6)

See Also

SumTools[Hypergeometric]

SumTools[Hypergeometric][EfficientRepresentation]

SumTools[Hypergeometric][MultiplicativeDecomposition]

SumTools[Hypergeometric][RationalCanonicalForm]

SumTools[Hypergeometric][SumDecomposition]