 ExtendSeries - Maple Help

QDifferenceEquations

 ExtendSeries
 extend a series solution of a q-difference equation to higher degree Calling Sequence ExtendSeries(sol, data, deg, dataname) Parameters

 sol - formal series solution of the system that is the result of an invocation of SeriesSolution or ExtendSeries data - special data needed to extend the series, returned by SeriesSolution or ExtendSeries deg - positive integer; formal degree of the initial terms to extend to dataname - (optional) name; if given the name is set to special data needed to extend the series found to higher degree with further invocation of QDifferenceEquations[ExtendSeries] Description

 • 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. Examples

 > $\mathrm{with}\left(\mathrm{QDifferenceEquations}\right):$
 > $\mathrm{eq}≔y\left(qx\right)-\left(1+{x}^{2}\right)y\left(x\right)$
 ${\mathrm{eq}}{≔}{y}{}\left({q}{}{x}\right){-}\left({{x}}^{{2}}{+}{1}\right){}{y}{}\left({x}\right)$ (1)
 > $\mathrm{var}≔y\left(x\right)$
 ${\mathrm{var}}{≔}{y}{}\left({x}\right)$ (2)
 > $\mathrm{sol}≔\mathrm{SeriesSolution}\left(\mathrm{eq},\mathrm{var},\varnothing ,\mathrm{output}=\mathrm{basis}\left[\mathrm{_C}\right],'\mathrm{data}'\right)$
 ${\mathrm{sol}}{≔}{{\mathrm{_C}}}_{{1}}{+}{\mathrm{O}}{}\left({x}\right)$ (3)
 > $\mathrm{sol}≔\mathrm{ExtendSeries}\left(\mathrm{sol},\mathrm{data},4,'\mathrm{data}'\right)$
 ${\mathrm{sol}}{≔}{{\mathrm{_C}}}_{{1}}{+}\frac{{{\mathrm{_C}}}_{{1}}{}{{x}}^{{2}}}{{{q}}^{{2}}{-}{1}}{+}\frac{{{\mathrm{_C}}}_{{1}}{}{{x}}^{{4}}}{\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{4}}{-}{1}\right)}{+}{\mathrm{O}}{}\left({{x}}^{{5}}\right)$ (4)
 > $\mathrm{eq}≔\left(1-{q}^{10}-\left(q-{q}^{10}\right)x\right)y\left({q}^{2}x\right)-\left(1-{q}^{20}-\left({q}^{2}-{q}^{20}\right)x\right)y\left(qx\right)+{q}^{10}\left(1-{q}^{10}-\left({q}^{2}-{q}^{11}\right)x\right)y\left(x\right)=\left({q}^{21}-{q}^{20}-{q}^{12}+{q}^{10}+{q}^{2}-q\right)x+{x}^{20}$
 ${\mathrm{eq}}{≔}\left({1}{-}{{q}}^{{10}}{-}\left({-}{{q}}^{{10}}{+}{q}\right){}{x}\right){}{y}{}\left({{q}}^{{2}}{}{x}\right){-}\left({1}{-}{{q}}^{{20}}{-}\left({-}{{q}}^{{20}}{+}{{q}}^{{2}}\right){}{x}\right){}{y}{}\left({q}{}{x}\right){+}{{q}}^{{10}}{}\left({1}{-}{{q}}^{{10}}{-}\left({-}{{q}}^{{11}}{+}{{q}}^{{2}}\right){}{x}\right){}{y}{}\left({x}\right){=}\left({{q}}^{{21}}{-}{{q}}^{{20}}{-}{{q}}^{{12}}{+}{{q}}^{{10}}{+}{{q}}^{{2}}{-}{q}\right){}{x}{+}{{x}}^{{20}}$ (5)
 > $\mathrm{var}≔y\left(x\right)$
 ${\mathrm{var}}{≔}{y}{}\left({x}\right)$ (6)
 > $\mathrm{sol}≔\mathrm{SeriesSolution}\left(\mathrm{eq},\mathrm{var},\mathrm{output}=\mathrm{onesol},'\mathrm{data}'\right)$
 ${\mathrm{sol}}{≔}{1}{+}{{x}}^{{10}}{+}{\mathrm{O}}{}\left({{x}}^{{11}}\right)$ (7)
 > $\mathrm{sol}≔\mathrm{ExtendSeries}\left(\mathrm{sol},\mathrm{data},20\right)$
 ${\mathrm{sol}}{≔}{{x}}^{{10}}{+}{1}{-}\frac{{{x}}^{{20}}}{{{q}}^{{10}}{}\left({{q}}^{{40}}{-}{2}{}{{q}}^{{30}}{+}{2}{}{{q}}^{{10}}{-}{1}\right)}{+}{\mathrm{O}}{}\left({{x}}^{{21}}\right)$ (8)