QDifferenceEquations
QMultiplicativeDecomposition
construct the four minimal multiplicative decompositions of a q-hypergeometric term
Calling Sequence
Parameters
Description
Examples
References
QMultiplicativeDecomposition[1](H, q, n, k)
QMultiplicativeDecomposition[2](H, q, n, k)
QMultiplicativeDecomposition[3](H, q, n, k)
QMultiplicativeDecomposition[4](H, q, n, k)
H
-
q-hypergeometric term in q^n
q
name used as the parameter q, usually q
n
variable
k
name
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=Wqn∏k=n0n−1Fqk 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.
withQDifferenceEquations:
H≔Productqk+q2qk+1qk+q5−q3qk+q4−q2q3qk+q2−1q12qk+q2−1qk+q5qk+q42q4qk+1qk+q2−1q2qk+q2−1,k=0..n−1
H≔∏k=0n−1qk+q2qk+1qk+q5−q3qk+q4−q2q3qk+q2−1q12qk+q2−1qk+q5qk+q42q4qk+1qk+q2−1q2qk+q2−1
QMultiplicativeDecomposition1H,q,n,k
1q10nqn+q2−1q22q3+qn2q4+qn2q+qnq2+qnqn+q2−1q11qn+q2−1q10qn+q2−1q9qn+q2−1q8qn+q2−1q7qn+q2−1q6qn+q2−1q5qn+q2−1q4qn+q2−1q3qn+q2−1qq2+qn−1∏k=0n−1qk+q5−q3qk+q4−q2qk+q5qk+1q41+q2−1q22q3+12q4+12q+1q2+11+q2−1q111+q2−1q101+q2−1q91+q2−1q81+q2−1q71+q2−1q61+q2−1q51+q2−1q41+q2−1q31+q2−1qq2
QMultiplicativeDecomposition2H,q,n,k
2q3−q+12q4−q2+121+1q31+1q21+1q1+q2−1qq2q5−q3+1q18nq3+qnq4+qn∏k=0n−1qk+q2−1q3qk+q2−1q12qk+q4qk+q5q3+1q4+1q3+qn−q2qn+q4−q22qn+1q3qn+1q2qn+1qqn+1qn+q2−1qq2+qn−1qn+q5−q3
QMultiplicativeDecomposition3H,q,n,k
q3−q+1q4−q2+1q4nq3+qn2q4+qn2q+qnq2+qnqn+q2−1q2∏k=0n−1qk+q5−q3qk+q2−1q12qk+q5qk+1q4q3+12q4+12q+1q2+11+q2−1q2q3+qn−qqn+q4−q2
QMultiplicativeDecomposition4H,q,n,k
2q3−q+1q4−q2+11+1q31+1q21+1qq12nqn+q2−1q2q3+qnq4+qn∏k=0n−1qk+q5−q3qk+q2−1q12qk+q5qk+q41+q2−1q2q3+1q4+1q3+qn−qqn+q4−q2qn+1q3qn+1q2qn+1qqn+1
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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