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SumTools[Hypergeometric]

  

RationalCanonicalForm

  

construct four rational canonical forms of a rational function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

RationalCanonicalForm[1](F, n)

RationalCanonicalForm[2](F, n)

RationalCanonicalForm[3](F, n)

RationalCanonicalForm[4](F, n)

Parameters

F

-

rational function of n

n

-

variable

Description

• 

Let F be a rational function of n over a field K of characteristic 0. The RationalCanonicalForm[i](F,n) calling sequence constructs the ith rational canonical forms for F, i=1,2,3,4.

  

If the RationalCanonicalForm command is called without an index, the first rational canonical form is constructed.

• 

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

1. 

F=zrEuvvsu.

2. 

gcdr,Eks=1 for all integers k.

3. 

gcdr,u.Ev=1, gcds,Eu.v=1.

  

Note: E is the automorphism of K(n) defined by EFn=Fn+1.

• 

The five-tuple z,r,s,u,v that satisfies the three conditions is a strict rational normal form for F. The rational functions zrs and uv are called the kernel and the shell of an RNFF, respectively.

• 

Let φ=z,r,s,u,v be any RNF 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 Fn=pnEGnqnGn where p, q are polynomials in n, and G is a rational function of n, then degreerdegreep and degreesdegreeq.

  

If i=1 then degreev is minimal.

  

If i=2 then degreeu is minimal.

  

If i=3 then degreeu+degreev is minimal, and under this condition, degreev is minimal.

  

If i=4 then degreeu+degreev is minimal, and under this condition, degreeu is minimal.

Examples

withSumTools[Hypergeometric]:

νnn+2n4+2n3+2n+2+2n+11+2

νnn+2n4+2n3+2n+2+2n+11+2

(1)

den3n22n+6n+12n1+2n+1+2

den3n22n+6n+12n1+2n+1+2

(2)

Fνde

Fnn+2n4+2n3+2n+2+2n+11+2n3n22n+6n+12n1+2n+1+2

(3)

z1,r1,s1,u1,v1RationalCanonicalForm[1]F,n

z1,r1,s1,u1,v11,n4+2n3+2,n3n+6n+12,n+1+22n12n22n+1nn+10+2n+9+2n+8+2n+7+2n+6+2n+5+2n+4+2n+3+2n+2+2n+2n1+2,1

(4)

z2,r2,s2,u2,v2RationalCanonicalForm[2]F,n

z2,r2,s2,u2,v21,n+2+2n+11+2,n3n22,1,n2+22n3+22n+52n+42n+32n+22n+2n1+2n4+2n+11n+10n+9n+8n+7n+6n+1n

(5)

z3,r3,s3,u3,v3RationalCanonicalForm[3]F,n

z3,r3,s3,u3,v31,n4+2n+11+2,n3n+6n+12,n12n22n+1nn+1+2,n2+2n3+2

(6)

z4,r4,s4,u4,v4RationalCanonicalForm[4]F,n

z4,r4,s4,u4,v41,n4+2n+11+2,n3n2n+12,n1n2n+1+2,n+5n+4n+3n+2n2+2n3+2

(7)

Check the result from RationalCanonicalForm[1].

Condition 1 is satisfied.

evalbF=normalz1r1s1subsn=n+1,u1v1u1v1

true

(8)

Condition 2 is satisfied.

LREtools[dispersion]r1,s1,n,LREtools[dispersion]s1,r1,n

FAIL,FAIL

(9)

Condition 3 is satisfied.

gcdr1,u1subsn=n+1,v1,gcds1,subsn=n+1,u1v1

1,1

(10)

Degrees of the kernel:

degreer1,n,degreer2,n,degreer3,n,degreer4,n

2,2,2,2

(11)

degrees1,n,degrees2,n,degrees3,n,degrees4,n

3,3,3,3

(12)

The degree of v1 is minimal.

degreev1,n,degreev2,n,degreev3,n,degreev4,n

0,23,2,6

(13)

The degree of u2 is minimal.

degreeu1,n,degreeu2,n,degreeu3,n,degreeu4,n

19,0,7,3

(14)

For i=3,4, the degree of the shell is minimal.

degreeu1,n+degreev1,n,degreeu2,n+degreev2,n,degreeu3,n+degreev3,n,degreeu4,n+degreev4,n

19,23,9,9

(15)

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.

  

Abramov, S.A., and Petkovsek, M. "Canonical representations of hypergeometric terms." Proc. FPSAC'2001, pp. 1-10. 2001.

See Also

evalb

LREtools[dispersion]

subs

SumTools[Hypergeometric]

SumTools[Hypergeometric][EfficientRepresentation]

SumTools[Hypergeometric][MultiplicativeDecomposition]

SumTools[Hypergeometric][PolynomialNormalForm]

SumTools[Hypergeometric][SumDecomposition]