QSimpComb - Maple Help

QDifferenceEquations

 QSimpComb
 simplification of expressions involving q-hypergeometric terms
 QSimplify
 simplification of expressions involving q-hypergeometric terms

 Calling Sequence QSimpComb(f) QSimplify(f)

Parameters

 f - algebraic expression

Description

 • The commands QSimpComb and QSimplify are for simplification of expressions involving q-hypergeometric terms. For a function $f\left({q}^{k}\right)$, the main use of QSimpComb is for detecting if $f\left({q}^{k}\right)$ is a q-hypergeometric term in ${q}^{k}$. That is, if $\frac{f\left({q}^{k+1}\right)}{f\left({q}^{k}\right)}$ is a rational function in ${q}^{k}$ (see IsQHypergeometricTerm). If the result is not a rational function, QSimplify returns in general a more compact answer.
 • This implementation is mainly based on the implementation by H. Boeing, W. Koepf. See the Reference Section.

Examples

 > $\mathrm{with}\left(\mathrm{QDifferenceEquations}\right):$
 > $H≔\frac{{\left(\frac{{\left({q}^{2}-1\right)}^{2}}{{q}^{6}}\right)}^{n}\mathrm{QPochhammer}\left(\frac{1}{-{q}^{5}+{q}^{3}},q,n\right)\mathrm{QPochhammer}\left(\frac{1}{-{q}^{4}+{q}^{2}},q,n\right)\mathrm{QPochhammer}\left(-\frac{1{q}^{3}}{{q}^{2}-1},q,n\right)\mathrm{QPochhammer}\left(-\frac{1}{{q}^{2}},q,n\right)\mathrm{QPochhammer}\left(-\frac{1{q}^{12}}{{q}^{2}-1},q,n\right)\mathrm{QPochhammer}\left(-1,q,n\right)}{\mathrm{QPochhammer}\left(-\frac{1{q}^{2}}{{q}^{2}-1},q,n\right)\mathrm{QPochhammer}\left(-\frac{1}{{q}^{5}},q,n\right){\mathrm{QPochhammer}\left(-\frac{1}{{q}^{4}},q,n\right)}^{2}\mathrm{QPochhammer}\left(-{q}^{4},q,n\right)\mathrm{QPochhammer}\left(\frac{1}{-{q}^{2}+1},q,n\right)}$
 ${H}{≔}\frac{{\left(\frac{{\left({{q}}^{{2}}{-}{1}\right)}^{{2}}}{{{q}}^{{6}}}\right)}^{{n}}{}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{5}}{+}{{q}}^{{3}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{4}}{+}{{q}}^{{2}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{3}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{2}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{12}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-1}{,}{q}{,}{n}\right)}{{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{2}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{5}}}{,}{q}{,}{n}\right){}{{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{4}}}{,}{q}{,}{n}\right)}^{{2}}{}{\mathrm{QPochhammer}}{}\left({-}{{q}}^{{4}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{2}}{+}{1}}{,}{q}{,}{n}\right)}$ (1)

Apply QSimpComb to the consecutive ratio $\frac{H\left(n+1\right)}{H\left(n\right)}$. If the result is a rational function in ${q}^{n}$, then H is a q-hypergeometric term.

 > $\mathrm{QSimpComb}\left(\frac{\mathrm{subs}\left(n=n+1,H\right)}{H}\right)$
 $\frac{\left({{q}}^{{5}}{-}{{q}}^{{3}}{+}{{q}}^{{n}}\right){}\left({{q}}^{{2}}{+}{{q}}^{{n}}\right){}\left({1}{+}{{q}}^{{n}}\right){}\left({{q}}^{{n}}{}{{q}}^{{12}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{n}}{}{{q}}^{{3}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{4}}{-}{{q}}^{{2}}{+}{{q}}^{{n}}\right)}{\left({{q}}^{{2}}{+}{{q}}^{{n}}{-}{1}\right){}\left({{q}}^{{2}}{}{{q}}^{{n}}{+}{{q}}^{{2}}{-}{1}\right){}{\left({{q}}^{{4}}{+}{{q}}^{{n}}\right)}^{{2}}{}\left({1}{+}{{q}}^{{n}}{}{{q}}^{{4}}\right){}\left({{q}}^{{5}}{+}{{q}}^{{n}}\right)}$ (2)
 > $\mathrm{IsQHypergeometricTerm}\left(H,n,{q}^{n}=N\right)$
 ${\mathrm{true}}$ (3)
 > $f≔\mathrm{QPochhammer}\left(a{q}^{-kn},q,n\right)-\frac{\mathrm{QPochhammer}\left(\frac{q}{a},q,kn\right){\left(-a\right)}^{n}{q}^{\mathrm{binomial}\left(n,2\right)-k{n}^{2}}}{\mathrm{QPochhammer}\left(\frac{q}{a},q,kn-n\right)}$
 ${f}{≔}{\mathrm{QPochhammer}}{}\left({a}{}{{q}}^{{-}{k}{}{n}}{,}{q}{,}{n}\right){-}\frac{{\mathrm{QPochhammer}}{}\left(\frac{{q}}{{a}}{,}{q}{,}{k}{}{n}\right){}{\left({-}{a}\right)}^{{n}}{}{{q}}^{\left(\genfrac{}{}{0}{}{{n}}{{2}}\right){-}{k}{}{{n}}^{{2}}}}{{\mathrm{QPochhammer}}{}\left(\frac{{q}}{{a}}{,}{q}{,}{k}{}{n}{-}{n}\right)}$ (4)
 > $\mathrm{QSimplify}\left(f\right)$
 ${0}$ (5)
 > $f≔\frac{1\mathrm{QPochhammer}\left(a,{q}^{2},n\right)\mathrm{QPochhammer}\left(aq,{q}^{2},n\right)}{\mathrm{QPochhammer}\left(a,q,2n\right)}$
 ${f}{≔}\frac{{\mathrm{QPochhammer}}{}\left({a}{,}{{q}}^{{2}}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({q}{}{a}{,}{{q}}^{{2}}{,}{n}\right)}{{\mathrm{QPochhammer}}{}\left({a}{,}{q}{,}{2}{}{n}\right)}$ (6)
 > $\mathrm{QSimpComb}\left(f\right)$
 $\frac{{\mathrm{QPochhammer}}{}\left({a}{,}{{q}}^{{2}}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({q}{}{a}{,}{{q}}^{{2}}{,}{n}\right)}{{\mathrm{QPochhammer}}{}\left({a}{,}{q}{,}{2}{}{n}\right)}$ (7)
 > $\mathrm{QSimplify}\left(f\right)$
 ${1}$ (8)

References

 Boeing, H., and Koepf, W. "Algorithms for q-hypergeometric summation in computer algebra." Journal of Symbolic Computation. Vol. 11. (1999): 1-23.