The ApplyOperator function applies the operator $L$ to the series $S$ using the elementary operations for series: differentiation, derivative representation, and multiplication by a polynomial.

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

$\mathrm{S1}≔\mathrm{Create}\left(2^n\,\mathrm{LaguerreL}\left(n\,1\,x\right)\right)$

$\mathrm{S1}≔\sum _{n=0}^{\mathrm{\infty }}{2}^n\mathrm{LaguerreL}\left(n\,1\,x\right)$

$\mathrm{R1}≔\mathrm{ApplyOperator}\left(\left[x^2{\mathrm{dx}}^2-7x\mathrm{dx}+3\,\left[\mathrm{dx}\,x\right]\right]\,\mathrm{S1}\right)$

$\mathrm{R1}≔139\mathrm{LaguerreL}\left(0\,3\,x\right)+332\mathrm{LaguerreL}\left(1\,3\,x\right)+\left(\sum _{n=2}^{\mathrm{\infty }}\left(-192^{n+3}n+32^{n+2}n-82^{n}n+152^{n+1}n-122^{n+3}-332^{n+2}+32^n+222^{n+1}+2^{n+4}{n}^2-42^{n+3}{n}^2+62^{n+2}{n}^2+2^n{n}^2-42^{n+1}{n}^2+92^{n+4}n+202^{n+4}\right)\mathrm{LaguerreL}\left(n\,3\,x\right)\right)$

$\mathrm{SimplifyCoefficients}\left(\mathrm{R1}\,\mathrm{simplify}\right)$

$139\mathrm{LaguerreL}\left(0\,3\,x\right)+332\mathrm{LaguerreL}\left(1\,3\,x\right)+\left(\sum _{n=2}^{\mathrm{\infty }}{2}^n\left(n^2+26n+139\right)\mathrm{LaguerreL}\left(n\,3\,x\right)\right)$

$\mathrm{S3}≔\mathrm{Create}\left(a\left(n\,m\right)\,\mathrm{LaguerreL}\left(n\,2\,x\right)\,\mathrm{LaguerreL}\left(m\,3\,y\right)\right)$

$\mathrm{S3}≔\sum _{m=0}^{\mathrm{\infty }}\left(\sum _{n=0}^{\mathrm{\infty }}a\left(n\,m\right)\mathrm{LaguerreL}\left(n\,2\,x\right)\right)\mathrm{LaguerreL}\left(m\,3\,y\right)$

$R≔\mathrm{ApplyOperator}\left(\left[x\mathrm{dx}+y\mathrm{dy}\,\left[\mathrm{dx}\,x\right]\,\left[\mathrm{dy}\,y\right]\right]\,\mathrm{S3}\right)$

$R≔\sum _{m=0}^{\mathrm{\infty }}\left(\sum _{n=0}^{\mathrm{\infty }}\left(na\left(n\,m\right)+\left(-2n-4\right)a\left(n+1\,m\right)+\left(n+4\right)a\left(n+2\,m\right)+ma\left(n\,m\right)+\left(-2m-5\right)a\left(n\,m+1\right)+\left(m+5\right)a\left(n\,m+2\right)\right)\mathrm{LaguerreL}\left(n\,2\,x\right)\right)\mathrm{LaguerreL}\left(m\,3\,y\right)$

$\mathrm{SimplifyCoefficients}\left(R\,\mathrm{collect}\,a\right)$

$\sum _{m=0}^{\mathrm{\infty }}\left(\sum _{n=0}^{\mathrm{\infty }}\left(\left(n+m\right)a\left(n\,m\right)+\left(-2n-4\right)a\left(n+1\,m\right)+\left(n+4\right)a\left(n+2\,m\right)+\left(-2m-5\right)a\left(n\,m+1\right)+\left(m+5\right)a\left(n\,m+2\right)\right)\mathrm{LaguerreL}\left(n\,2\,x\right)\right)\mathrm{LaguerreL}\left(m\,3\,y\right)$

$\mathrm{S5}≔\mathrm{Create}\left(\left\{\frac{1}{n+1}\,\left[1=7\right]\right\}\,\mathrm{LaguerreL}\left(n\,1\,x\right)\right)$

$\mathrm{S5}≔7\mathrm{LaguerreL}\left(1\,1\,x\right)+\left(\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{LaguerreL}\left(n\,1\,x\right)}{n+1}\right)$

$\mathrm{R2}≔\mathrm{ApplyOperator}\left(\left[\left(1+x\right)d+a\,\left[d\,x\right]\right]\,\mathrm{S5}\right)$

$\mathrm{R2}≔\left(-\frac{13a}{2}-29\right)\mathrm{LaguerreL}\left(0\,2\,x\right)+\left(7a+7\right)\mathrm{LaguerreL}\left(1\,2\,x\right)+\left(\sum _{n=1}^{\mathrm{\infty }}\left(\frac{a}{\left(n+1\right)\left(n+2\right)}-\frac{1}{n+2}-\frac{1}{\left(n+1\right)\left(n+2\right)}\right)\mathrm{LaguerreL}\left(n\,2\,x\right)\right)$

$\mathrm{SimplifyCoefficients}\left(\mathrm{R2}\,\mathrm{simplify}\right)$

$\left(-\frac{13a}{2}-29\right)\mathrm{LaguerreL}\left(0\,2\,x\right)+\left(7a+7\right)\mathrm{LaguerreL}\left(1\,2\,x\right)+\left(\sum _{n=1}^{\mathrm{\infty }}\frac{\left(a-n-2\right)\mathrm{LaguerreL}\left(n\,2\,x\right)}{\left(n+1\right)\left(n+2\right)}\right)$