The gfun[`rec+rec`](rec1, rec2, u(n)) command returns a termwise sum of two holonomic recurrences, rec1 and rec2.

If $a\left(n\right)$ and $b\left(n\right)$ are the sequences defined by rec1 and rec2 respectively, the gfun[`rec+rec`] function returns a recurrence for $a\left(n\right)+b\left(n\right)$.

The gfun[`rec*rec`](rec1, rec2, u(n)) command returns a termwise product of two holonomic recurrences, rec1 and rec2.

If $a\left(n\right)$ and $b\left(n\right)$ are the sequences defined by rec1 and rec2 respectively, the gfun[`rec*rec`] function returns a recurrence for $a\left(n\right)b\left(n\right)$.

The gfun[cauchyproduct](rec1, rec2, u(n)) command returns the Cauchy product of the two holonomic recurrences, rec1 and rec2.

If $a\left(n\right)$ and $b\left(n\right)$ are the sequences defined by rec1 and rec2 respectively, the gfun[cauchyproduct] function returns a recurrence for their Cauchy product or convolution $c\left(n\right)=\sum _{i=0}^{n}a\left(i\right)b\left(n-i\right)$.

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

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

$\mathrm{rec2}≔u\left(n+1\right)=2u\left(n\right)\:$

$\mathrm{`rec+rec`}\left(\mathrm{rec1}\,\mathrm{rec2}\,u\left(n\right)\right)$

$\left\{\left(2n^2+2n\right)u\left(n\right)+\left(-n^2-3n+2\right)u\left(n+1\right)+\left(n-1\right)u\left(n+2\right)\,u\left(0\right)={\mathrm{\_t}}_2+{\mathrm{\_C}}_0\,u\left(1\right)={\mathrm{\_t}}_2+2{\mathrm{\_C}}_0\,u\left(2\right)=2{\mathrm{\_t}}_2+4{\mathrm{\_C}}_0\,u\left(3\right)=6{\mathrm{\_t}}_2+8{\mathrm{\_C}}_0\right\}$

$\mathrm{`rec*rec`}\left(\mathrm{rec1}\,\mathrm{rec2}\,u\left(n\right)\right)$

$\left(-2n-2\right)u\left(n\right)+u\left(n+1\right)$

$\mathrm{cauchyproduct}\left(\mathrm{rec1}\,\mathrm{rec2}\,u\left(n\right)\right)$

$\left\{\left(2n+4\right)u\left(n\right)+\left(-4-n\right)u\left(n+1\right)+u\left(n+2\right)\,u\left(0\right)={\mathrm{\_C}}_0\,u\left(1\right)=3{\mathrm{\_C}}_0\right\}$