QRationalCanonicalForm - Maple Help
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QRationalCanonicalForm

  

construct four q-rational canonical forms of a rational function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

QRationalCanonicalForm[1](F, q, n)

QRationalCanonicalForm[2](F, q, n)

QRationalCanonicalForm[3](F, q, n)

QRationalCanonicalForm[4](F, q, n)

Parameters

F

-

rational function of n

q

-

name used as the parameter q, usually q

n

-

variable

Description

• 

Let F be a rational function of n over a field K of characteristic 0, q is a nonzero element of K which is not a root of unity. The QRationalCanonicalForm[i](F,q,n) command constructs the th rational canonical form for F, .

  

If QRationalCanonicalForm is called without an index, the first q-rational canonical form is constructed.

• 

The output is a sequence of 5 elements , called , where z is an element of K, and  are monic polynomials over K such that:

1. 

, .

2. 

 for all integers .

3. 

, .

4. 

, .

  

Note: Q is the automorphism of K(n) defined by .

• 

The five-tuple  that satisfies the four conditions is a strict q-rational normal form for F. The rational function  and  are called the kernel and the shell of the , respectively.

• 

Let  be any qRNF of a rational function F. Then the degrees of the polynomials r and s are unique, and have minimal possible values in the sense that if  where p, q are polynomials in n, and G is a rational function of n, then  and .

• 

Additionally, if  then  is minimal; if  then  is minimal; if  then  is minimal, and under this condition,  is minimal; if  then  is minimal, and under this condition,  is minimal.

Examples

(1)

(2)

(3)

(4)

(5)

Check the result from QRationalCanonicalForm[2].

Condition 1 is satisfied.

(6)

Condition 2 is satisfied.

(7)

Condition 3 is satisfied.

(8)

Condition 4 is satisfied.

(9)

Degrees of the kernel:

(10)

(11)

The degree of v1 is minimal:

(12)

The degree of u2 is minimal:

(13)

For , the degree of the shell is minimal:

(14)

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[QDispersion]

QDifferenceEquations[QEfficientRepresentation]

QDifferenceEquations[QMultiplicativeDecomposition]

QDifferenceEquations[QPolynomialNormalForm]

 


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