We construct a generalized expression for the normal ordering of (a + a†)m for integral values of m and use the result to study the quantum anharmonic oscillator problem in the Heisenberg approach. In particular, we derive generalized expressions for energy eigenvalues and frequency shifts for the Hamiltonian H = ((x2)/2) + ((2)/2) + (λ/m)xm. We also derive a closed-form first-order multiscale perturbation theoretic operator solution of this Hamiltonian with a view to generalize some recent results of Bender and Bettencourt (1996 Phys. Rev. Lett. 77 4114, 1996 Phys. Rev. D 54 1710).