QDifferenceEquations - Maple Programming Help

Home : Support : Online Help : Mathematics : Factorization and Solving Equations : QDifferenceEquations : QDifferenceEquations/QPolynomialNormalForm

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

 > $\mathrm{with}\left(\mathrm{QDifferenceEquations}\right):$
 > $F≔\frac{\left(n-1\right)\left({q}^{3}n-1\right)}{\left(qn-1\right)\left({q}^{4}n-1\right)}$
 ${F}{≔}\frac{\left({n}{-}{1}\right){}\left({n}{}{{q}}^{{3}}{-}{1}\right)}{\left({n}{}{q}{-}{1}\right){}\left({n}{}{{q}}^{{4}}{-}{1}\right)}$ (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.

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

Condition 2 is satisfied.

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

Condition 3 is satisfied.

 > $\genfrac{}{}{0}{}{c}{\phantom{n=0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{c}}{n=0}\ne 0$
 $\frac{{1}}{{{q}}^{{3}}}{\ne }{0}$ (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.