The IsSolution(sol,eq,var) calling sequence returns true if eq is a linear q-difference equation with polynomial coefficients (or a system of such equations) and sol is its solution. Otherwise, false is returned for q-hypergeometric solution (that is, if 'certificates' option is specified) and the degree of the result of substitution is returned for other types of solutions to be checked.

The IsSolution command substitutes sol for the function variables and checks the result.

The AreSameSolution(sol, sol1, x) calling sequence returns true if the solutions sol and sol1 are equivalent. Otherwise, false is returned.

The command checks that the numbers of linear independent solutions in sol and sol1 are the same, as well as the degrees in x.

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

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

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

$\mathrm{sol1}≔\mathrm{PolynomialSolution}\left(\mathrm{eq}\,y\left(x\right)\,\varnothing \,\mathrm{output}=\mathrm{basis}\left[\mathrm{\_K}\right]\right)$

$\mathrm{sol1}≔{\mathrm{\_K}}_1{x}^{10}+{\mathrm{\_K}}_{2}x-{\mathrm{\_K}}_2+1$

$\mathrm{IsSolution}\left(\mathrm{sol1}\,\mathrm{eq}\,y\left(x\right)\right)$

$\mathrm{true}$

$\mathrm{sol2}≔\mathrm{sol1}-x^5$

$\mathrm{sol2}≔{\mathrm{\_K}}_1{x}^{10}-x^5+{\mathrm{\_K}}_{2}x-{\mathrm{\_K}}_2+1$

$\mathrm{IsSolution}\left(\mathrm{sol2}\,\mathrm{eq}\,y\left(x\right)\right)$

$6$

$\mathrm{sol3}≔\mathrm{sol1}-x^{10}$

$\mathrm{sol3}≔{\mathrm{\_K}}_1{x}^{10}-x^{10}+{\mathrm{\_K}}_{2}x-{\mathrm{\_K}}_2+1$

$\mathrm{IsSolution}\left(\mathrm{sol3}\,\mathrm{eq}\,y\left(x\right)\right)$

$\mathrm{true}$

$\mathrm{AreSameSolution}\left(\mathrm{sol1}\,\mathrm{sol3}\,x\right)$

$\mathrm{true}$

$\mathrm{sol4}≔\mathrm{sol1}-\mathrm{\_K}\left[1\right]x^{10}$

$\mathrm{sol4}≔{\mathrm{\_K}}_{2}x-{\mathrm{\_K}}_2+1$

$\mathrm{IsSolution}\left(\mathrm{sol4}\,\mathrm{eq}\,y\left(x\right)\right)$

$\mathrm{true}$

$\mathrm{AreSameSolution}\left(\mathrm{sol1}\,\mathrm{sol4}\,x\right)$

$\mathrm{false}$