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SumTools[Hypergeometric]

  

MultiplicativeDecomposition

  

construct the four minimal multiplicative decompositions of a hypergeometric term

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

MultiplicativeDecomposition[1](H, n, k)

MultiplicativeDecomposition[2](H, n, k)

MultiplicativeDecomposition[3](H, n, k)

MultiplicativeDecomposition[4](H, n, k)

Parameters

H

-

hypergeometric term of n

n

-

variable

k

-

name

Description

• 

Let H be a hypergeometric term of n. The MultiplicativeDecomposition[i](H,n,k) calling sequence constructs the ith minimal multiplicative decomposition of H of the form Hn=Wnk=n0n1Fk where Wn,Fn are rational functions of n, degreenumerFn and degreedenomFn have minimal possible values, for i=1,2,3,4.

  

If i=1 then degreedenomW is minimal.

  

If i=2 then degreenumerW is minimal.

  

If i=3 then degreenumerW+degreedenomW is minimal, and degreedenomW is minimal.

  

If i=4 then degreenumerW+degreedenomW is minimal, and degreenumerW is minimal.

  

If the MultiplicativeDecomposition command is called without an index, the first minimal multiplicative decomposition is constructed.

Examples

withSumToolsHypergeometric:

H24k=4n1kk+2k4+212k3+212k+2+212k+11+212k3k22k+6k+12k1+212k+1+212

H24k=4n1kk+2k4+2k3+2k+2+2k+11+2k3k22k+6k+12k1+2k+1+2

(1)

MultiplicativeDecomposition1H,n,k

n+1+22n12n22n+1nn+10+2n+9+2n+8+2n+7+2n+6+2n+5+2n+4+2n+3+2n+2+2n+2n1+2k=4n1k4+2k3+2k+12k+6k3305+2214+213+212+211+210+29+28+27+26+24+23+2

(2)

MultiplicativeDecomposition2H,n,k

158176291553280002+221+224+23+22k=4n1k+2+2k+11+2k3k22n2+22n3+22n+52n+42n+32n+22n+2n1+2n4+2n+11n+10n+9n+8n+7n+6n+1n

(3)

MultiplicativeDecomposition3H,n,k

2+21+2n+1+2n12n22n+1nk=4n1k4+2k+11+2k+12k+6k3305+2n2+2n3+2

(4)

MultiplicativeDecomposition4H,n,k

120962+21+2n+1+2n1n2k=4n1k4+2k+11+2k3k2k+125+2n2+2n3+2n+5n+4n+3n+2

(5)

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][EfficientRepresentation]

SumTools[Hypergeometric][RationalCanonicalForm]

SumTools[Hypergeometric][SumDecomposition]