Connections between Laguerre polynomials through a third-order differential operator transformation

Let \(\{l^{(\alpha)}_n\}_{n\geq 0}\), (\(\alpha\neq-m, m\geq1\)), be the monic orthogonal sequence of Laguerre polynomials. We define a new differential operator, \(\mathscr{L}^{+}_{\alpha}\), that raises the degree and also the parameter of \(l^{(\alpha)}_n(x)\). More precisely, \(\mathscr{L}^{+}_{\alpha}l^{(\alpha)}_n(x)=l^{(\alpha+1)}_{n+1}(x), n\geq0\). As an illustration, we give some properties related to this operator and some other operators in the literature.