The invborel(expr, a(n)) command computes the inverse Borel transform of a generating function.

If $a\left(n\right),n=0..\mathrm{\infty }$ is the sequence of numbers defined by the recurrence expr, the invborel function computes the recurrence for the numbers $n!a\left(n\right)$.

If t is specified as 'diffeq', expr is considered a linear differential equation with polynomial coefficients.  The function returns a linear differential equation satisfied by its inverse Borel transform.

Laplace can be used as a synonym of invborel, that is, the Laplace(expr, a(n)) command can be used instead.

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

$\mathrm{rec}≔\left\{a\left(0\right)=0\,a\left(1\right)=1\,a\left(n\right)=na\left(n-1\right)+a\left(n-2\right)\right\}\:$

$\mathrm{deq}≔\mathrm{invborel}\left(\mathrm{rec}\,a\left(n\right)\right)$

$\mathrm{deq}≔\left\{\left(-n^2-3n-2\right)a\left(n\right)+\left(-n^2-4n-4\right)a\left(n+1\right)+a\left(n+2\right)\,a\left(0\right)=0\,a\left(1\right)=1\right\}$

$\mathrm{rec2}≔\mathrm{rectodiffeq}\left(\mathrm{deq}\,a\left(n\right)\,f\left(t\right)\right)$

$\mathrm{rec2}≔\left(-2t^2-t+1\right)f\left(t\right)+\left(-4t^3-3t^2\right)\left(\frac{\ⅆ}{\ⅆt}f\left(t\right)\right)+\left(-t^4-t^3\right)\left(\frac{{\ⅆ}^2}{\ⅆt^2}f\left(t\right)\right)-t$

$\mathrm{deq2}≔\mathrm{borel}\left(\mathrm{rec2}\,f\left(t\right)\,'\mathrm{diffeq}'\right)$

$\mathrm{deq2}≔\left(-t^2-t+1\right)f\left(t\right)-\left(\frac{\ⅆ}{\ⅆt}f\left(t\right)\right)t^2-t$

$\mathrm{diffeqtorec}\left(\mathrm{deq2}\,f\left(t\right)\,a\left(n\right)\right)$

$\left\{-a\left(n\right)+\left(-n-2\right)a\left(n+1\right)+a\left(n+2\right)\,a\left(0\right)=0\,a\left(1\right)=1\right\}$