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: $F\=\frac{zaQ\left(c\right)}{bc}\.$  $\gcd \left(a,Q^{k\left(b\right)}\right)=1\mathrm{for\; all}\mathrm{non}-\mathrm{negative\; integers}k\.$ $c\left(0\right)\ne 0.$ $\gcd \left(a\,c\right)=1,\gcd \left(b\,Q\left(c\right)\right)=1.$

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

$\mathrm{with}\left(\mathrm{QDifferenceEquations}\right)\:$

$F≔\frac{\left(n-1\right)\left(q^{3}n-1\right)}{\left(qn-1\right)\left(q^{4}n-1\right)}$

$F≔\frac{\left(n-1\right)\left(q^{3}n-1\right)}{\left(qn-1\right)\left(q^{4}n-1\right)}$

$z,a,b,c≔\mathrm{QPolynomialNormalForm}\left(F\,q\,n\right)$

$z,a,b,c≔\frac{1}{q^4},n-1,n-\frac{1}{q^4},\left(n-\frac{1}{q^2}\right)\left(n-\frac{1}{q}\right)$

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

$0$

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

$\mathrm{FAIL}$

$\mathrm{eval}\left(c\,n=0\right)\ne 0$

$\frac{1}{q^3}\ne 0$

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

$1,1$

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.