Differential Equations, Special Functions, Laplace Transform by Differential Calculus

A formula changing the operator f ( A)g( B ) where AB  BA  Iˆ into a sum of operators g ( k )( B ) f ( k )( A ) / k! is proved. Thank to this relation between operators a new and rapid method for resolutions of differential equations is exposed in details. It is seen to be useful also for obtaining the differential operators that transform monomials into Hermite, Laguerre, associated Laguerre, Gegenbauer, Chebyshev polynomials and for getting quasi all their main properties in a very concise manner. Is proposed also the differential representation of the Laplace transform permitting the differential calculus to prove consicely its properties.