Coherent state and normal ordering method for transiting Hermite polynomials to Laguerre polynomials

By virtue of the coherent state representation and the operator ordering method we find a new approach for transiting Hermite polynomials to Laguerre polynomials. We also derive the new reciprocal relation of Laguerre polynomials \(\sum\limits_{n = 0} {( - 1)^n \left( {_n^l } \right)L_n (x) = \tfrac{{x^l }} {{l!}}}\), and its application in deriving the sum rule of the Wingner function of Fock states is demonstrated. Some new expansion identities about the operator Laguerre polynomial are also derived. This opens a new route of deriving mathematical polynomials formulas by virtute of the quantum mechanical representations and operator ordering technique.