The ExtendSeries command returns the initial terms of the formal series solution sol extended to the specified formal degree deg.

Additionally, if a name given in the dataname parameter, the command sets the name to special data needed to extend the series found to higher degree with further invocation of QDifferenceEquations[ExtendSeries] command.

The ExtendSeries command solves the problem with a single q-difference equation and also with a system of such equations. In the latter case the command invokes LinearFunctionalSystems[ExtendSeries] in order to find solutions.

The solution is a series expansion in x, corresponding to var. The order term (for example $\mathrm{O}\left(x^6\right)$) is the last term in the series. For the system case the solution is a list of such series expansions.

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

$\mathrm{eq}≔y\left(qx\right)-\left(1+x^2\right)y\left(x\right)$

$\mathrm{eq}≔y\left(qx\right)-\left(x^2+1\right)y\left(x\right)$

$\mathrm{var}≔y\left(x\right)$

$\mathrm{var}≔y\left(x\right)$

$\mathrm{sol}≔\mathrm{SeriesSolution}\left(\mathrm{eq}\,\mathrm{var}\,\varnothing \,\mathrm{output}=\mathrm{basis}\left[\mathrm{\_C}\right]\,'\mathrm{data}'\right)$

$\mathrm{sol}≔{\mathrm{\_C}}_1+\mathrm{O}\left(x\right)$

$\mathrm{sol}≔\mathrm{ExtendSeries}\left(\mathrm{sol}\,\mathrm{data}\,4\,'\mathrm{data}'\right)$

$\mathrm{sol}≔{\mathrm{\_C}}_1+\frac{{\mathrm{\_C}}_1{x}^2}{q^2-1}+\frac{{\mathrm{\_C}}_1{x}^4}{\left(q^2-1\right)\left(q^4-1\right)}+\mathrm{O}\left(x^5\right)$

$\mathrm{eq}≔\left(1-q^{10}-\left(q-q^{10}\right)x\right)y\left(q^{2}x\right)-\left(1-q^{20}-\left(q^2-q^{20}\right)x\right)y\left(qx\right)+q^{10}\left(1-q^{10}-\left(q^2-q^{11}\right)x\right)y\left(x\right)=\left(q^{21}-q^{20}-q^{12}+q^{10}+q^2-q\right)x+x^{20}$

$\mathrm{eq}≔\left(1-q^{10}-\left(-q^{10}+q\right)x\right)y\left(q^{2}x\right)-\left(1-q^{20}-\left(-q^{20}+q^2\right)x\right)y\left(qx\right)+q^{10}\left(1-q^{10}-\left(-q^{11}+q^2\right)x\right)y\left(x\right)=\left(q^{21}-q^{20}-q^{12}+q^{10}+q^2-q\right)x+x^{20}$

$\mathrm{var}≔y\left(x\right)$

$\mathrm{var}≔y\left(x\right)$

$\mathrm{sol}≔\mathrm{SeriesSolution}\left(\mathrm{eq}\,\mathrm{var}\,\mathrm{output}=\mathrm{onesol}\,'\mathrm{data}'\right)$

$\mathrm{sol}≔1+x^{10}+\mathrm{O}\left(x^{11}\right)$

$\mathrm{sol}≔\mathrm{ExtendSeries}\left(\mathrm{sol}\,\mathrm{data}\,20\right)$

$\mathrm{sol}≔x^{10}+1-\frac{x^{20}}{q^{10}\left(q^{40}-2q^{30}+2q^{10}-1\right)}+\mathrm{O}\left(x^{21}\right)$