The authors establish the following bilateral generating function of the modified Laguerre polynomials and claims that it is the extension of their previous result (can be obtained by putting \(k=0).\)
If \(G(x,w)=\sum^{\infty}_{n=0}a_ nL_{a,b,m,n+k}(x)w^ n\), then \[ (1-wb)^{-m-k} \exp (-wax/(1-wb))G(x/(1-wb),wz/(1- wb))=\sum^{\infty}_{n=0}\sigma_ n(z)L_{a,b,m,n+k}(x)w^ n, \] where, \(\sigma_ n(z)=\sum^{n}_{p=0}a_ p\left( \begin{matrix} n+k\\ p+k\end{matrix} \right)z^ p\). To prove it they take the help of the linear partial differential operator \[ bxy\frac{\partial}{\partial x}+by^ 2\frac{\partial}{\partial y}+y\{b(m+k)-ax\}. \]