The IsSolution(sol, r, eq, var) command returns true if eq is a linear difference equation with polynomial coefficients depending on a hypergeometric term and sol is its solution. Otherwise, false is returned.

The IsSolution function substitutes sol for the function variable and checks the result.

The AreSameSolution(sol, r, sol1, r1, n) command returns true if the solutions sol and sol1 are equivalent. Otherwise, false is returned.

The function transforms sol and sol1 to have the same term and checks that the number of linear independent solutions and degrees of corresponding elements in sol and sol1 are the same.

$\mathrm{with}\left(\mathrm{LREtools}\left[\mathrm{HypergeometricTerm}\right]\right)\:$

$\mathrm{eq}≔y\left(n+2\right)-\left(n!+n\right)y\left(n+1\right)+n\left(n!-1\right)y\left(n\right)$

$\mathrm{eq}≔y\left(n+2\right)-\left(n!+n\right)y\left(n+1\right)+n\left(n!-1\right)y\left(n\right)$

$\mathrm{sol},r≔\mathrm{PolynomialSolution}\left(\mathrm{eq}\,y\left(n\right)\right)$

$\mathrm{sol},r≔\frac{t{\mathrm{\_C}}_1}{n},\left[t\,n+1\right]$

$\mathrm{IsSolution}\left(\mathrm{sol}\,r\,\mathrm{eq}\,y\left(n\right)\right)$

$\mathrm{true}$

$\mathrm{sol1}≔t\mathrm{\_C}\left[1\right]$

$\mathrm{sol1}≔t{\mathrm{\_C}}_1$

$\mathrm{r1}≔\left[t\,n\right]$

$\mathrm{r1}≔\left[t\,n\right]$

$\mathrm{IsSolution}\left(\mathrm{sol1}\,\mathrm{r1}\,\mathrm{eq}\,y\left(n\right)\right)$

$\mathrm{true}$

$\mathrm{AreSameSolution}\left(\mathrm{sol1}\,\mathrm{r1}\,\mathrm{sol}\,r\,n\right)$

$\mathrm{true}$

$\mathrm{eq}≔\left(2^{n}n!+n^2\right)z\left(n+1\right)-\left(2n{2}^{n}n!+22^{n}n!+n^2+2n+1\right)z\left(n\right)$

$\mathrm{eq}≔\left(2^{n}n!+n^2\right)z\left(n+1\right)-\left(2n{2}^{n}n!+22^{n}n!+n^2+2n+1\right)z\left(n\right)$

$\mathrm{sol},r≔\mathrm{PolynomialSolution}\left(\mathrm{eq}\,z\left(n\right)\right)$

$\mathrm{sol},r≔{\mathrm{\_C}}_1{n}^2+t{\mathrm{\_C}}_1,\left[t\,2n+2\right]$

$\mathrm{IsSolution}\left(\mathrm{sol}\,r\,\mathrm{eq}\,z\left(n\right)\right)$

$\mathrm{true}$

$\mathrm{sol1}≔\mathrm{\_C}\left[1\right]n^2+tn\left(n-1\right)\left(n-2\right)\mathrm{\_C}\left[1\right]$

$\mathrm{sol1}≔{\mathrm{\_C}}_1{n}^2+tn\left(n-1\right)\left(n-2\right){\mathrm{\_C}}_1$

$\mathrm{r1}≔\left[t\,2n-4\right]$

$\mathrm{r1}≔\left[t\,2n-4\right]$

$\mathrm{IsSolution}\left(\mathrm{sol1}\,\mathrm{r1}\,\mathrm{eq}\,z\left(n\right)\right)$

$\mathrm{true}$

$\mathrm{AreSameSolution}\left(\mathrm{sol1}\,\mathrm{r1}\,\mathrm{sol}\,r\,n\right)$

$\mathrm{true}$

$\mathrm{sol2}≔n^2+tn\left(n-1\right)\left(n-2\right)$

$\mathrm{sol2}≔n^2+tn\left(n-1\right)\left(n-2\right)$

$\mathrm{IsSolution}\left(\mathrm{sol2}\,\mathrm{r1}\,\mathrm{eq}\,z\left(n\right)\right)$

$\mathrm{true}$

$\mathrm{AreSameSolution}\left(\mathrm{sol2}\,\mathrm{r1}\,\mathrm{sol}\,r\,n\right)$

$\mathrm{false}$