The poltorec(P, listrec, list_unknowns, u(n)) command returns the recurrence satisfied by the polynomial P.

If $\mathrm{u1}\left(n\right)$, $\mathrm{u2}\left(n\right)$, ... are holonomic sequence solutions of listrec[1], listrec[2], ..., the poltorec function returns a linear recurrence equation satisfied by $P\left(n,\mathrm{u1}\left(n\right),...\right)$.

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

$\mathrm{rec1}≔\left\{\mathrm{u1}\left(0\right)=1\,\mathrm{u1}\left(n+1\right)=\left(n+1\right)\mathrm{u1}\left(n\right)\right\}\:$

$\mathrm{rec2}≔\left\{\mathrm{u2}\left(0\right)=1\,\mathrm{u2}\left(1\right)=1\,\mathrm{u2}\left(n+2\right)=2\mathrm{u2}\left(n+1\right)-3n\mathrm{u2}\left(n\right)\right\}\:$

$\mathrm{poltorec}\left({\mathrm{u1}\left(n\right)}^2+2\mathrm{u1}\left(n\right)\mathrm{u2}\left(n\right)\,\left[\mathrm{rec1}\,\mathrm{rec2}\right]\,\left[\mathrm{u1}\left(n\right)\,\mathrm{u2}\left(n\right)\right]\,u\left(n\right)\right)$

$\left\{\left(-3n^7-39n^6-192n^5-462n^4-579n^3-363n^2-90n\right)u\left(n\right)+\left(5n^5+54n^4+209n^3+354n^2+254n+60\right)u\left(n+1\right)+\left(-n^4-12n^3-46n^2-62n-15\right)u\left(n+2\right)+\left(n^2+4n\right)u\left(n+3\right)\,u\left(0\right)=3\,u\left(1\right)=3\,u\left(2\right)=12\,u\left(3\right)=48\right\}$

$\mathrm{fib}≔\left\{F\left(0\right)=1\,F\left(1\right)=1\,F\left(n+2\right)=F\left(n+1\right)+F\left(n\right)\right\}\:$

$\mathrm{poltorec}\left(F\left(n+2\right)F\left(n\right)-{F\left(n+1\right)}^2\,\left[\mathrm{fib}\right]\,\left[F\left(n\right)\right]\,f\left(n\right)\right)$

$\left\{f\left(n+1\right)+f\left(n\right)\,f\left(0\right)=1\right\}$