RegularQPochhammerForm - Maple Help

QDifferenceEquations

 RegularQPochhammerForm
 construct the regular q-Pochhammer representation of a q-hypergeometric term

 Calling Sequence RegularQPochhammerForm(H, q, n)

Parameters

 H - q-hypergeometric term of n q - name used as the parameter q, usually q n - variable

Description

 • Let H be a q-hypergeometric term of q^n, R be the certificate of H, and n0 be an integer such that R has neither a pole nor a zero for all $\mathrm{n0}\le n$. Let R factor into linear factors

$R≔\frac{z\left(x-{a}_{1}\right)\mathrm{...}\left(x-{a}_{r}\right)}{\left(x-{b}_{1}\right)\mathrm{....}\left(x-{b}_{s}\right)}$

 The RegularQPochhammerForm(H,q,n) command returns the multiplicative decomposition of the form $H\left({q}^{\mathrm{n0}}\right)C{w}^{n-\mathrm{n0}}P\left(n\right)$ where

$P≔\frac{\mathrm{QPochhammer}\left(\frac{1}{{a}_{1}},q,n\right)\mathrm{...}\mathrm{QPochhammer}\left(\frac{1}{{a}_{r}},q,n\right)}{\mathrm{QPochhammer}\left(\frac{1}{{b}_{1}},q,n\right)\mathrm{....}\mathrm{QPochhammer}\left(\frac{1}{{b}_{s}},q,n\right)}$

$w≔\frac{{\left(-1\right)}^{r+s}z{a}_{1}\mathrm{...}{a}_{r}}{{b}_{1}\mathrm{....}{b}_{s}}$

$C≔\frac{{q}^{\left(\genfrac{}{}{0}{}{n}{2}\right)}\mathrm{QPochhammer}\left(\frac{1}{{b}_{1}},q,\mathrm{n0}\right)\mathrm{...}\mathrm{QPochhammer}\left(\frac{1}{{b}_{s}},q,\mathrm{n0}\right)}{{q}^{\left(\genfrac{}{}{0}{}{\mathrm{n0}}{2}\right)}\mathrm{QPochhammer}\left(\frac{1}{{a}_{1}},q,\mathrm{n0}\right)\mathrm{....}\mathrm{QPochhammer}\left(\frac{1}{{a}_{r}},q,\mathrm{n0}\right)}$

Examples

 > $\mathrm{with}\left(\mathrm{QDifferenceEquations}\right):$
 > $H≔\mathrm{Product}\left(\frac{\left({q}^{k}+{q}^{2}\right)\left({q}^{k}+1\right)\left({q}^{k}+{q}^{5}-{q}^{3}\right)\left({q}^{k}+{q}^{4}-{q}^{2}\right)\left({q}^{3}{q}^{k}+{q}^{2}-1\right)\left({q}^{12}{q}^{k}+{q}^{2}-1\right)}{\left({q}^{k}+{q}^{5}\right){\left({q}^{k}+{q}^{4}\right)}^{2}\left({q}^{4}{q}^{k}+1\right)\left({q}^{k}+{q}^{2}-1\right)\left({q}^{2}{q}^{k}+{q}^{2}-1\right)},k=0..n-1\right)$
 ${H}{≔}{\prod }_{{k}{=}{0}}^{{n}{-}{1}}{}\frac{\left({{q}}^{{k}}{+}{{q}}^{{2}}\right){}\left({{q}}^{{k}}{+}{1}\right){}\left({{q}}^{{k}}{+}{{q}}^{{5}}{-}{{q}}^{{3}}\right){}\left({{q}}^{{k}}{+}{{q}}^{{4}}{-}{{q}}^{{2}}\right){}\left({{q}}^{{3}}{}{{q}}^{{k}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{12}}{}{{q}}^{{k}}{+}{{q}}^{{2}}{-}{1}\right)}{\left({{q}}^{{k}}{+}{{q}}^{{5}}\right){}{\left({{q}}^{{k}}{+}{{q}}^{{4}}\right)}^{{2}}{}\left({{q}}^{{4}}{}{{q}}^{{k}}{+}{1}\right){}\left({{q}}^{{k}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{2}}{}{{q}}^{{k}}{+}{{q}}^{{2}}{-}{1}\right)}$ (1)
 > $\mathrm{RegularQPochhammerForm}\left(H,q,n\right)$
 $\frac{{\left(\frac{{\left({{q}}^{{2}}{-}{1}\right)}^{{2}}}{{{q}}^{{6}}}\right)}^{{n}}{}{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{12}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-1}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{3}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{4}}{+}{{q}}^{{2}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{2}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{5}}{+}{{q}}^{{3}}}{,}{q}{,}{n}\right)}{{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{2}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}{{q}}^{{4}}{,}{q}{,}{n}\right){}{{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{4}}}{,}{q}{,}{n}\right)}^{{2}}{}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{2}}{+}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{5}}}{,}{q}{,}{n}\right)}$ (2)