On some connection results between Laguerre polynomials via third-order
differential operator
Abstract
Let
$\{L^{(\alpha)}_n\}_{n\geq
0}$, ($\alpha\neq-m, \
m\geq1$), be the monic orthogonal sequence of Laguerre
polynomials. We give a new differential operator, denoted here
$\mathscr{L}^{+}_{\alpha}$,
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, then we give some connection results between Laguerre
polynomials via this new operator.