Connection formulas and representations of Laguerre polynomials in terms of the action of linear differential operators

In this paper, we introduce the notion of $\mathfrak{O}_{\varepsilon}$-classical orthogonal polynomials, where $\mathfrak{O}_{\varepsilon}:=\mathbb{I}+\varepsilon D$ ($\varepsilon\neq0$). It is shown that the scaled Laguerre polynomial sequence $\{a^{-n}L^{(\alpha)}_n(ax)\}_{n\geq0}$, where $a=-\varepsilon^{-1}$, is actually the only $\mathfrak{O}_{\varepsilon}$-classical sequence. As an illustration, we deal with some representations of Laguerre polynomials $L^{(0)}_n(x)$ in terms of the action of linear differential operators on the Laguerre polynomials $L^{(m)}_n(x)$. The inverse connection problem of expanding Laguerre polynomials $L^{(m)}_n(x)$ in terms of $L^{(0)}_n(x)$ is also considered. Furthermore, some connection formulas between the monomial basis $\{x^n\}_{n\geq0}$ and the shifted Laguerre basis $\{L^{(m)}_n(x+1)\}_{n\geq0}$ are deduced.