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QDifferenceEquations

  

QEfficientRepresentation

  

construct the four efficient representations of a q-hypergeometric term

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

QEfficientRepresentation[1](H, q, n)

QEfficientRepresentation[2](H, q, n)

QEfficientRepresentation[3](H, q, n)

QEfficientRepresentation[4](H, q, n)

Parameters

H

-

q-hypergeometric term in q^n

q

-

name used as the parameter q, usually q

n

-

variable

Description

• 

Let H be a q-hypergeometric term in qn. The QEfficientRepresentation[i](H,q,n) command constructs the ith efficient representation of H of the form Hn=CαnVqnQn where C, α are constant and Qn is a product of QPochhammer-function values and their reciprocals. Additionally,

1. 

Qn has the minimal number of factors,

2. 

Vqn is a rational function which is minimal in one sense or another, depending on the particular q-rational canonical form chosen to represent the certificate of Hqn.

• 

If i=1 then degreedenomV is minimal; if i=2 then degreenumerV is minimal; if i=3 then degreenumerV+degreedenomV is minimal, and under this condition, degreedenomV is minimal; if i=4 then degreenumerV+degreedenomV is minimal, and under this condition, degreenumerV is minimal.

• 

If QEfficientRepresentation is called without an index, the first efficient representation 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)

QEfficientRepresentation[1]H,q,n

q66qn+q21q22q3+qn2q4+qn2q+qnq2+qnqn+q21q11qn+q21q10qn+q21q9qn+q21q8qn+q21q7qn+q21q6qn+q21q5qn+q21q4qn+q21q3qn+q21qq2+qn1q212q6nQDifferenceEquations:-QPochhammer1q4+q2,q,nQDifferenceEquations:-QPochhammer1q5+q3,q,n2q212q3+12q4+12q+1q2+1q11+q21q10+q21q9+q21q8+q21q7+q21q6+q21q5+q21q4+q21q3+q21q2+q1QDifferenceEquations:-QPochhammer1q5,q,nQDifferenceEquations:-QPochhammerq4,q,n

(2)

QEfficientRepresentation[2]H,q,n

2q5q3+1q2+q1q+1q2+1q4q2+12q3q+12q3+qnq4+qnq212q6nQDifferenceEquations:-QPochhammerq3q21,q,nQDifferenceEquations:-QPochhammerq12q21,q,nq5q4+1q3+qnq2qn+q4q22qn+1q3qn+1q2qn+1qqn+1qn+q21qq2+qn1qn+q5q3QDifferenceEquations:-QPochhammer1q4,q,nQDifferenceEquations:-QPochhammer1q5,q,n

(3)

QEfficientRepresentation[3]H,q,n

q2q3q+1q4q2+1q3+qn2q4+qn2q+qnq2+qnqn+q21q2q212q6nQDifferenceEquations:-QPochhammerq12q21,q,nQDifferenceEquations:-QPochhammer1q5+q3,q,nq3+12q4+12q+1q2+12q21q3+qnqqn+q4q2QDifferenceEquations:-QPochhammer1q5,q,nQDifferenceEquations:-QPochhammerq4,q,n

(4)

QEfficientRepresentation[4]H,q,n

2q4q2+1q3q+1q+1q2+1q3+qnq4+qnqn+q21q2q212q6nQDifferenceEquations:-QPochhammerq12q21,q,nQDifferenceEquations:-QPochhammer1q5+q3,q,nq42q21q4+1qn+1q3qn+1q2qn+1qqn+1q3+qnqqn+q4q2QDifferenceEquations:-QPochhammer1q4,q,nQDifferenceEquations:-QPochhammer1q5,q,n

(5)

RegularQPochhammerFormH,q,n

q212q6nQDifferenceEquations:-QPochhammer1q4+q2,q,nQDifferenceEquations:-QPochhammerq3q21,q,nQDifferenceEquations:-QPochhammer1,q,nQDifferenceEquations:-QPochhammerq12q21,q,nQDifferenceEquations:-QPochhammer1q2,q,nQDifferenceEquations:-QPochhammer1q5+q3,q,nQDifferenceEquations:-QPochhammerq2q21,q,nQDifferenceEquations:-QPochhammerq4,q,nQDifferenceEquations:-QPochhammer1q4,q,n2QDifferenceEquations:-QPochhammer1q2+1,q,nQDifferenceEquations:-QPochhammer1q5,q,n

(6)

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[QMultiplicativeDecomposition]

QDifferenceEquations[QObjects]

QDifferenceEquations[QRationalCanonicalForm]

QDifferenceEquations[RegularQPochhammerForm]

 


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