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.

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.

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

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

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

$\mathrm{var}≔y\left(x\right)$

$\mathrm{var}≔y\left(x\right)$

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

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

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

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

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

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

$\mathrm{var}≔y\left(x\right)$

$\mathrm{var}≔y\left(x\right)$

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

$\mathrm{sol}≔{\mathrm{\_C}}_1+\left(-{\mathrm{\_C}}_1+1\right)x+{\mathrm{\_C}}_2{x}^{10}+\mathrm{O}\left(x^{11}\right)$

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

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