The PolynomialSolution(eq, var, term) command returns the polynomial solution of the linear difference equation eq. If such a solution does not exist, the function returns NULL.

The hypergeometric term in the linear difference equation is specified by a name, for example, t. The meaning of the term is defined by the parameter term. It can be specified directly in the form of an equation, for example, $t=n!$, or specified as a list consisting of the name of term variable and the consecutive term ratio, for example, $\left[t\,n+1\right]$.

If the third parameter is omitted, then the input equation can contain a hypergeometric term directly (not a name). In this case, the procedure extracts the term from the equation, transforms the equation to the form with a name representing a hypergeometric term, and then solves the transformed equation.

The term "polynomial solution" means a solution $y\left(x\right)$ in $Q\left(x\right)\left[t,t^{-1}\right]$ , that is, in the form $y=y_d{t}^d+...+y_g{t}^g$ where $d\le g$ and $y_d,...,y_g$ are in $Q\left(x\right)$.

The solution is the function, corresponding to var. The solution involves arbitrary constants of the form, for example, _c1 and _c2.

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

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

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

$\mathrm{PolynomialSolution}\left(\mathrm{eq}\,y\left(n\right)\,t=n!\right)$

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

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

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

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

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

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

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

$\mathrm{PolynomialSolution}\left(\mathrm{eq}\,z\left(n\right)\,t=2^{n}n!\right)$

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

$\mathrm{eq}≔45y\left(x\right)-9y\left(x\right)x-18y\left(x+3\right)+9y\left(x+3\right)x$

$\mathrm{eq}≔45y\left(x\right)-9y\left(x\right)x-18y\left(x+3\right)+9y\left(x+3\right)x$

$\mathrm{PolynomialSolution}\left(\mathrm{eq}\,y\left(x\right)\,\left[t\,\frac{9\cdot 1}{10-7x-8x^2}\right]\right)$

$\frac{{\mathrm{\_C}}_1}{x-5},\left[t\,\frac{9}{-8x^2-7x+10}\right]$

Bronstein, M. "On solutions of Linear Ordinary Difference Equations in their Coefficients Field." INRIA Research Report. No. 3797. November 1999.