Deriving new identities of Hermite-polynomial operators and Laguerre-polynomial operators by using representation theory and operator-ordering technique

By virtue of the linear combination of a†and a and the coordinate-momentum intermediate representation as well as operator-ordering technique we derive some new expansion identities about Hermite-polynomial operators and Laguerre-polynomial operators. We also find a new approach for transiting the Hermite polynomial operator to the associated Laguerre polynomial operator. The reciprocal formula of the associated Laguerre polynomial is also derived.