The PolynomialSolution(eq, var, term) command returns the universal denominator of the rational solution of the linear difference equation eq.

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 the term variable and the consecutive term ratio, for example, $\left[t\,n+1\right]$.

The term "rational solution" means a solution $y\left(x\right)$ in $Q\left(x\right)\left(t\right)$. (See PolynomialSolution for the meaning of "polynomial solution".) Here we use the term "denominator" which is q in $Q\left(x\right)\left[t\right]$ to mean that $qy$ is in $Q\left(x\right)\left[t,t^{-1}\right]$ .

The search for a rational solution is based on finding a universal denominator which is u in $Q\left(x\right)\left[t\right]$ such that $uy$ is in $Q\left(x\right)\left[t,t^{-1}\right]$ for any rational solution y. By replacing y with $\frac{Y}{u}$ in the given equation, we reduce the problem to searching for a polynomial solution.

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

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

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

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

$1+t$

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

$\mathrm{eq}≔y\left(n+2\right)n^2{t}^2-y\left(n\right)n^{3}t-y\left(n\right)n^2{t}^2+y\left(n+2\right)n^{2}t+3y\left(n+2\right)n{t}^2-3y\left(n\right)n^{2}t-3y\left(n\right)n{t}^2+4y\left(n+2\right)nt+2y\left(n+2\right)t^2-2y\left(n\right)nt-2y\left(n\right)t^2+y\left(n+2\right)n+4y\left(n+2\right)t-y\left(n\right)n-y\left(n\right)t+2y\left(n+2\right)$

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

$n+t$

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

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

$\mathrm{UniversalDenominator}\left(\mathrm{eq}\,y\left(n\right)\,t=2^n\right)$

$n+t$

Abramov, S.A., and Bronstein, M. "Hypergeometric dispersion and the orbit problem." Proc. ISSAC 2000.

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