SeriesSolution - Maple Help

QDifferenceEquations

 SeriesSolution
 return a series solution of a q-difference equation

 Calling Sequence SeriesSolution(eq, var, inits, output=type, dataname)

Parameters

 eq - q-difference equation or a list of such equations (for the system case) var - function variable to solve for, such as $y\left(x\right)$, or a list of such function variables (for the system case) inits - set of initial conditions output=type - (optional) where type is one of basis[C] or onesol and "C" is a name. The words output, basis and onesol must be used literally. dataname - (optional) name; if given the name is set to special data needed to extend the series found to higher degree with QDifferenceEquations[ExtendSeries]

Description

 • The SeriesSolution command returns the series solution of the given linear q-difference equation with polynomial coefficients. If such a solution does not exist, then NULL is returned.
 • Additionally, if a name given in the dataname parameter, then the command sets the name to special data needed to extend the series found to higher degree with QDifferenceEquations[ExtendSeries] command.
 • The SeriesSolution 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[SeriesSolution] 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.
 • For a single q-difference equation the function computes some initial terms of the series. The number of the terms is determined in such a way that the process of computing the successive terms does not lead to new arbitrary constants. The successive terms can be computed with the QDifferenceEquations[ExtendSeries].
 • The parameter q in a scalar q-difference equation can be either a name or a rational number.

Output options

 • Optionally, you can specify output=basis, output=basis[var], output=onesol, output=gensol, or output=anysol.
 output=basis[C]
 The output is provided as a single algebraic expression that is a $C$-linear combination of the independent solutions plus any particular solution for the inhomogeneous case. The independent solutions will have indexed coefficients of the form ${C}_{0},{C}_{1},...,{C}_{n}$, where var is as provided in the output=basis[C] option.
 output=onesol
 This specifies that only a single solution be provided as output.
 • The inits argument is in fact ignored and allowed in the calling sequence for compatibility with the other solvers in the package.

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)
 > $\mathrm{sol}≔\mathrm{QDifferenceEquations}\left['\mathrm{SeriesSolution}'\right]\left(\mathrm{eq},\mathrm{var},\varnothing ,\mathrm{output}=\mathrm{basis}\left[\mathrm{_K}\right]\right)$
 ${\mathrm{sol}}{≔}{{\mathrm{_K}}}_{{1}}{+}{\mathrm{O}}{}\left({x}\right)$ (3)
 > $\mathrm{sol}≔\mathrm{QDifferenceEquations}\left['\mathrm{SeriesSolution}'\right]\left(\mathrm{eq},\mathrm{var},\varnothing ,\mathrm{output}=\mathrm{onesol}\right)$
 ${\mathrm{sol}}{≔}{1}{+}{\mathrm{O}}{}\left({x}\right)$ (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)
 > $\mathrm{sol}≔\mathrm{QDifferenceEquations}\left['\mathrm{SeriesSolution}'\right]\left(\mathrm{eq},\mathrm{var},\varnothing ,\mathrm{output}=\mathrm{basis}\left[\mathrm{_C}\right]\right)$
 ${\mathrm{sol}}{≔}{{\mathrm{_C}}}_{{1}}{+}\left({-}{{\mathrm{_C}}}_{{1}}{+}{1}\right){}{x}{+}{{\mathrm{_C}}}_{{2}}{}{{x}}^{{10}}{+}{\mathrm{O}}{}\left({{x}}^{{11}}\right)$ (7)
 > $\mathrm{sol}≔\mathrm{QDifferenceEquations}\left['\mathrm{SeriesSolution}'\right]\left(\mathrm{eq},\mathrm{var},\varnothing ,\mathrm{output}=\mathrm{onesol}\right)$
 ${\mathrm{sol}}{≔}{1}{+}{{x}}^{{10}}{+}{\mathrm{O}}{}\left({{x}}^{{11}}\right)$ (8)