The agfmomentsolve function accepts a set of multivariate generating functions. It attempts to solve the system of generating functions where the equations are differentiated num times with respect to each variable aside from size. It then evaluates the equations  with each nonsize variable set to 1.

The functions are returned as univariate functions.  Thus, it is important that the first variable marks size. The agfmomentsolve function is useful for finding averages and higher moments.

The modified equations are returned by using the third argument, neweqns. If it is specified, it is assigned to the system of equations.

If the solving computation is intensive, you can interrupt it and recover the equations. This allows you to intervene with some manual manipulation or specialized solving methods.

The agfmomentsolve function relies on solutions obtained by agfmomentsolve(eqns, num-j) where $j=1..\mathrm{num}$. Thus, if these cannot be solved, fail is returned.

For information on how to write combstruct grammar specifications, see combstruct and combstruct[specification].

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

$\mathrm{eqns0}≔\mathrm{agfeqns}\left(\left\{T=\mathrm{Prod}\left(Z\,\mathrm{Set}\left(T\right)\right)\right\}\,\left\{h\left(T\right)=\mathrm{Prod}\left(0\,\mathrm{Set}\left(h\left(T\right)\right)+1\right)\right\}\,\mathrm{labeled}\,z\,\left[\left[u\,h\right]\right]\right)$

$\mathrm{eqns0}≔\left[T\left(z\,u\right)=z{\ⅇ}^{T\left(z\,u\right)}u\,Z\left(z\,u\right)=zu\right]$

$\mathrm{eqns1}≔\mathrm{gfeqns}\left(\left\{E=\mathrm{{\rm E}}\,T=\mathrm{Prod}\left(Z\,\mathrm{Prod}\left(E\,\mathrm{Set}\left(T\right)\right)\right)\right\}\,\mathrm{labeled}\,z\,\left[\left[u\,E\right]\right]\right)$

$\mathrm{eqns1}≔\left[E\left(z\,u\right)=u\,T\left(z\,u\right)=z{\ⅇ}^{T\left(z\,u\right)}u\,Z\left(z\,u\right)=z\right]$

$\mathrm{agfmomentsolve}\left(\mathrm{eqns0}\,0\right)$

$\left\{T\left(z\right)=-\mathrm{LambertW}\left(-z\right)\,Z\left(z\right)=z\right\}$

$\mathrm{agfmomentsolve}\left(\mathrm{eqns0}\,1\,\mathrm{new}\right)$

$\left\{T_1\left(z\right)=-\frac{{\ⅇ}^{-\mathrm{LambertW}\left(-z\right)}}{z{\ⅇ}^{-\mathrm{LambertW}\left(-z\right)}-1}\,T_2\left(z\right)=-\frac{z{\ⅇ}^{-\mathrm{LambertW}\left(-z\right)}}{z{\ⅇ}^{-\mathrm{LambertW}\left(-z\right)}-1}\,Z_1\left(z\right)=1\,Z_2\left(z\right)=z\right\}$

$\mathrm{new}$

$\left[T_1\left(z\right)={\ⅇ}^{-\mathrm{LambertW}\left(-z\right)}+z{T}_1\left(z\right){\ⅇ}^{-\mathrm{LambertW}\left(-z\right)}\,T_2\left(z\right)=z{T}_2\left(z\right){\ⅇ}^{-\mathrm{LambertW}\left(-z\right)}+z{\ⅇ}^{-\mathrm{LambertW}\left(-z\right)}\,Z_1\left(z\right)=1\,Z_2\left(z\right)=z\right]$

$\mathrm{sol}≔\mathrm{agfmomentsolve}\left(\mathrm{eqns0}\,2\right)\:$

$\mathrm{subs}\left(\mathrm{sol}\,T\left[2,2\right]\left(z\right)\right)$

$\frac{{\left({\ⅇ}^{-\mathrm{LambertW}\left(-z\right)}\right)}^2{z}^2\left(z{\ⅇ}^{-\mathrm{LambertW}\left(-z\right)}-2\right)}{{\left({\ⅇ}^{-\mathrm{LambertW}\left(-z\right)}\right)}^3{z}^3-3{\left({\ⅇ}^{-\mathrm{LambertW}\left(-z\right)}\right)}^2{z}^2+3z{\ⅇ}^{-\mathrm{LambertW}\left(-z\right)}-1}$