Quantum mechanical operator Touchard polynomials studied by virtue of operators' normal ordering and Weyl ordering

In this paper, we propose quantum mechanical operator formalism for Touchard polynomials whose generating function is Tn(x)=partial derivative n partial derivative lambda ne(e lambda-1)x|lambda=0. That is replacing e lambda x by e lambda a dagger a, where a dagger a is the number operator, and using the method of integration within ordered product we find that (a dagger a)n is just the normal ordering form :Tn(a dagger a):. Then by virtue of the Weyl ordering form of quantum mechanical operator, we also introduce a new special polynomial whose generating function is Gn(x)=partial derivative n partial derivative lambda n2e lambda+1exp[2xe lambda-1e lambda+1]|lambda=0. With use of the Weyl ordering form of e lambda a dagger a, we prove (a dagger a)n=:Tn(a dagger a):=::Gn(a dagger a)::, where :::: denotes Weyl ordering. It seems that quantum mechancal operator formalism presents a new and simpler approach for generalizing Touchard polynomial theory.