construct the q-polynomial normal form of a rational function - Maple Help

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QDifferenceEquations[QPolynomialNormalForm] - construct the q-polynomial normal form of a rational function

Calling Sequence

QPolynomialNormalForm(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 QPolynomialNormalForm(F,q,n) command constructs the q-polynomial normal form for F.

• 

The output is a sequence of 4 elements z,a,b,c where z is an element of K, and a,b,c are monic polynomials over K such that: F=zaQcbc.  gcda,Qkb=1for allnonnegative integersk. c00. gcda,c=1,gcdb,Qc=1.

  

Note: Q is the automorphism of K(n) defined by {Q(F(n)) = F(q*n)}.

Examples

withQDifferenceEquations:

F:=n1q3n1qn1q4n1

F:=n1nq31nq1nq41

(1)

z,a,b,c:=QPolynomialNormalFormF,q,n

z,a,b,c:=1q4,n1,n1q4,n1q2n1q

(2)

Check the results.

Condition 1 is satisfied.

normalFzabsubsn=qn,cc

0

(3)

Condition 2 is satisfied.

QDispersionb,a,q,n

FAIL

(4)

Condition 3 is satisfied.

cn=0|cn=00

1q30

(5)

Condition 4 is satisfied.

gcdexa,c,n,gcdexb,subsn=qn,c,n

1,1

(6)

See Also

QDifferenceEquations[QDispersion], QDifferenceEquations[QRationalCanonicalForm]

References

  

Abramov, S.A., and Petkovsek, M. "Finding all q-hypergeometric solutions of q-difference equations." Proc. FPSAC '95, Univ.de Marne-la-Vall'ee, Noisy-le-Grand, pp. 1-10. 1995.

  

Koornwinder, T.H. "On Zeilberger's algorithm and its q-analogue: a rigorous description." J. Comput. Appl. Math. Vol. 48. (1993): 91-111.


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