The SubstituteTerm(sol, term, n) returns the solution with a directly substituted hypergeometric term instead of a name representing it in the solution.

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

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

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

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

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

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

$\mathrm{SubstituteTerm}\left(\mathrm{sol}\,r\,n\right)$

$\mathrm{\Gamma }\left(n\right){\mathrm{\_C}}_1$

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

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

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

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

$\mathrm{SubstituteTerm}\left(\mathrm{sol}\,r\,n\right)$

${\mathrm{\_C}}_1\left(n^2+2^n\mathrm{\Gamma }\left(n+1\right)\right)$

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