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QDifferenceEquations

  

QMultiplicativeDecomposition

  

construct the four minimal multiplicative decompositions of a q-hypergeometric term

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

QMultiplicativeDecomposition[1](H, q, n, k)

QMultiplicativeDecomposition[2](H, q, n, k)

QMultiplicativeDecomposition[3](H, q, n, k)

QMultiplicativeDecomposition[4](H, q, n, k)

Parameters

H

-

q-hypergeometric term in q^n

q

-

name used as the parameter q, usually q

n

-

variable

k

-

name

Description

• 

Let H be a q-hypergeometric term in q^n. The QMultiplicativeDecomposition[i](H,q,n,k) command constructs the ith minimal multiplicative decomposition of H of the form Hqn=Wqnk=n0n1Fqk where Wqn,Fqn are rational functions of q^n, degreenumerFqn and degreedenomFqn have minimal possible values, for i=1,2,3,4.

• 

Additionally, if i=1 then degreedenomW is minimal; if i=2 then degreenumerW is minimal; if i=3 then degreenumerW+degreedenomW is minimal, and under this condition, degreedenomW is minimal; if i=4 then degreenumerW+degreedenomW is minimal, and under this condition, degreenumerW is minimal.

  

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

Examples

withQDifferenceEquations:

Hk=0n1qk+q2qk+1qk+q5q3qk+q4q2q3qk+q21q12qk+q21qk+q5qk+q42q4qk+1qk+q21q2qk+q21

H:=k=0n1qk+q2qk+1qk+q5q3qk+q4q2q3qk+q21q12qk+q21qk+q5qk+q42q4qk+1qk+q21q2qk+q21

(1)

QMultiplicativeDecomposition[1]H,q,n,k

1q10nqn+q21q22q3+qn2q4+qn2q+qnq2+qnqn+q21q11qn+q21q10qn+q21q9qn+q21q8qn+q21q7qn+q21q6qn+q21q5qn+q21q4qn+q21q3qn+q21qq2+qn1k=0n1qk+q5q3qk+q4q2qk+q5qk+1q41+q21q22q3+12q4+12q+1q2+11+q21q111+q21q101+q21q91+q21q81+q21q71+q21q61+q21q51+q21q41+q21q31+q21qq2

(2)

QMultiplicativeDecomposition[2]H,q,n,k

2q3q+12q4q2+121+1q31+1q21+1q1+q21qq2q5q3+1q18nq3+qnq4+qnk=0n1qk+q21q3qk+q21q12qk+q4qk+q5q3+1q4+1q3+qnq2qn+q4q22qn+1q3qn+1q2qn+1qqn+1qn+q21qq2+qn1qn+q5q3

(3)

QMultiplicativeDecomposition[3]H,q,n,k

q3q+1q4q2+1q4nqn+q21q2q3+qn2q4+qn2q+qnq2+qnk=0n1qk+q5q3qk+q21q12qk+q5qk+1q41+q21q2q3+12q4+12q+1q2+1q3+qnqqn+q4q2

(4)

QMultiplicativeDecomposition[4]H,q,n,k

21+1q31+1q21+1qq3q+1q4q2+1q12nq3+qnq4+qnqn+q21q2k=0n1qk+q5q3qk+q21q12qk+q5qk+q4q3+1q4+11+q21q2qn+1q3qn+1q2qn+1qqn+1q3+qnqqn+q4q2

(5)

References

  

Abramov, S.A.; Le, H.Q.; and Petkovsek, M. "Efficient Representations of (q-)Hypergeometric Terms and the Assignment Problem." Submitted.

  

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

QDifferenceEquations[QEfficientRepresentation]

QDifferenceEquations[QObjects]

QDifferenceEquations[QRationalCanonicalForm]

 


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