QPolynomialNormalForm - 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: $F=\frac{zaQ\left(c\right)}{bc}.$  $\mathrm{gcd}\left(a,{Q}^{k\left(b\right)}\right)=1\mathrm{for all}\mathrm{non}-\mathrm{negative integers}k.$ $c\left(0\right)\ne 0.$ $\mathrm{gcd}\left(a,c\right)=1,\mathrm{gcd}\left(b,Q\left(c\right)\right)=1.$
 Note: Q is the automorphism of K(n) defined by {Q(F(n)) = F(q*n)}.

Examples

 > $\mathrm{with}\left(\mathrm{QDifferenceEquations}\right):$
 > $F≔\frac{\left(n-1\right)\left({q}^{3}n-1\right)}{\left(qn-1\right)\left({q}^{4}n-1\right)}$
 ${F}{≔}\frac{\left({n}{-}{1}\right){}\left({{q}}^{{3}}{}{n}{-}{1}\right)}{\left({q}{}{n}{-}{1}\right){}\left({{q}}^{{4}}{}{n}{-}{1}\right)}$ (1)
 > $z,a,b,c≔\mathrm{QPolynomialNormalForm}\left(F,q,n\right)$
 ${z}{,}{a}{,}{b}{,}{c}{≔}\frac{{1}}{{{q}}^{{4}}}{,}{n}{-}{1}{,}{n}{-}\frac{{1}}{{{q}}^{{4}}}{,}\left({n}{-}\frac{{1}}{{{q}}^{{2}}}\right){}\left({n}{-}\frac{{1}}{{q}}\right)$ (2)

Check the results.

Condition 1 is satisfied.

 > $\mathrm{normal}\left(F-\frac{z\left(\frac{a}{b}\right)\mathrm{subs}\left(n=qn,c\right)}{c}\right)$
 ${0}$ (3)

Condition 2 is satisfied.

 > $\mathrm{QDispersion}\left(b,a,q,n\right)$
 ${\mathrm{FAIL}}$ (4)

Condition 3 is satisfied.

 > $\mathrm{eval}\left(c,n=0\right)\ne 0$
 $\frac{{1}}{{{q}}^{{3}}}{\ne }{0}$ (5)

Condition 4 is satisfied.

 > $\mathrm{gcdex}\left(a,c,n\right),\mathrm{gcdex}\left(b,\mathrm{subs}\left(n=qn,c\right),n\right)$
 ${1}{,}{1}$ (6)

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.