This routine returns the differential operator or differential equation satisfied by the generating function associated to the recurrence. This operator is returned as either a DESol operator or a differential equation (both possibly with initial conditions). No attempt is made to explicitly solve the result.

For the definition of the format of a problem, see the help page for LREtools[REcreate]. The problem can be an RESol structure resulting from a call to LREtools[REcreate].

The output option val can be either DESol which requests operator output (the default) or diff which requests the output in terms of a differential system.

Note: In some cases initial conditions are also needed, so the output=diff result is returned as a set. If no initial conditions are needed, the output is a single ODE.

Note: This command performs the same operation as gfun[rectodiffeq] and the inverse operation is provided by gfun[diffeqtorec].

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

$\mathrm{REtoDE}\left(a\left(n+2\right)-2a\left(n+1\right)+a\left(n\right)=0\,a\left(n\right)\,\varnothing \,f\left(z\right)\right)$

$\mathrm{DESol}\left(\left\{2f+\left(4z-4\right)\mathrm{D}\left(f\right)+\left(z^2-2z+1\right){\mathrm{D}}^{\left(2\right)}\left(f\right)\right\}\,\left\{f\right\}\right)$

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

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

$\mathrm{REtoDE}\left(\mathrm{rec}\,f\left(z\right)\,\mathrm{output}=\mathrm{diff}\right)$

$2f\left(z\right)+\left(4z-4\right)\left(\frac{\ⅆ}{\ⅆz}f\left(z\right)\right)+\left(z^2-2z+1\right)\left(\frac{{\ⅆ}^2}{\ⅆz^2}f\left(z\right)\right)$

$\mathrm{REtoDE}\left(2na\left(n+2\right)+\left(n^2+1\right)a\left(n+1\right)-{\left(n+1\right)}^{2}a\left(n\right)=0\,a\left(n\right)\,\varnothing \,f\left(z\right)\right)$

$\mathrm{DESol}\left(\left\{\left(-8z+8\right)f+\left(-40z^2-8\right)\mathrm{D}\left(f\right)+\left(-28z^3+8z^2+8z\right){\mathrm{D}}^{\left(2\right)}\left(f\right)+\left(-4z^4+4z^3\right){\mathrm{D}}^{\left(3\right)}\left(f\right)\right\}\,\left\{f\right\}\,\left\{f\left(0\right)=\mathrm{D}\left(f\right)\left(0\right)\right\}\right)$

$\mathrm{REtoDE}\left(2na\left(n+2\right)+\left(n^2+1\right)a\left(n+1\right)-{\left(n+1\right)}^{2}a\left(n\right)=0\,a\left(n\right)\,\varnothing \,f\left(z\right)\,\mathrm{output}=\mathrm{diff}\right)$

$\left\{\left(-8z+8\right)f\left(z\right)+\left(-40z^2-8\right)\left(\frac{\ⅆ}{\ⅆz}f\left(z\right)\right)+\left(-28z^3+8z^2+8z\right)\left(\frac{{\ⅆ}^2}{\ⅆz^2}f\left(z\right)\right)+\left(-4z^4+4z^3\right)\left(\frac{{\ⅆ}^3}{\ⅆz^3}f\left(z\right)\right)\,f\left(0\right)=\mathrm{D}\left(f\right)\left(0\right)\right\}$